Problems & Puzzles: Puzzles

Puzzle 1233 Sum of Primes such that...

Paolo Lava sent the following nice puzzle:

Find the smallest integer that can be expressed as the sum of k different primes, simultaneously for all k’s between 2 and n, with n >= 2. 

I found 5, 10, 21, 34, 45, 58, 81, 106, 129, 195, 201, 238, 285, 338, ... . Here below, the list of sums (of course, there are different ways to reach the same sums):

5 = 2 + 3;
10 = 3 + 7 = 2 + 3 + 5;
21 = 2 + 19 = 3 + 5 + 13 = 2 + 3 + 5 + 11;
34 = 3 + 31 = 2 + 3 + 29 = 3 + 5 + 7 + 19 = 2 + 3 + 5 + 7 + 17;
45 = 2 + 43 = 3 + 5 + 37 = 2 + 3 + 11 + 29 = 3 + 5 + 7 + 11 + 19 = 2 + 3 + 5 + 7 + 11 + 17;
58 = 5 + 53 = 2 + 13 + 43 = 3 + 5 + 7 + 43 = 2 + 3 + 5 + 7 + 41 = 3 + 5 + 7 + 11 + 13 + 19 = 2 + 3 + 5 + 7 + 11 + 13 + 17;
81 = 2 + 79 = 3 + 37 + 41 = 2 + 11 + 31 + 37 = 5 + 7 + 11 + 17 + 41 = 2 + 5 + 7 + 11 + 19 + 37 = 3 + 5 + 7 + 11 + 13 + 19 + 23 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23;
106 = 3 + 103 = 2 + 3 + 101 = 3 + 13 + 43 + 47 = 2 + 3 + 11 + 43 + 47 = 3 + 5 + 7 + 11 + 37 + 43 = 2 + 3 + 5 + 7 + 17 + 29 + 43 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 31 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29;
129 = 2 + 127 = 3 + 13 + 113 = 2 + 3 + 11 + 113 =  3 + 5 + 7 + 11 + 103 = 2 + 3 + 5 + 7 + 11 + 101 = 3 + 5 + 7 + 11 + 13 + 17 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 71 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29;
195 = 2 + 193 = 3 + 11 + 181 = 2 + 3 + 11 + 179 = 3 + 5 + 7 + 13 + 167 = 2 + 3 + 5 + 7 + 11 + 167 = 3 + 5 + 7 + 11 + 13 + 29 + 127 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 137 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 97 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 41 + 73 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37;
201 = 2 + 199 = 3 + 5 + 193 = 2 + 3 + 5 + 191 = 3 + 5 + 7 + 13 + 173 = 2 + 3 + 5 + 7 + 17 + 167 = 3 + 5 + 7 + 11 + 17 + 19 + 139 = 2 + 3 + 5 + 7 + 11 + 17 + 19 + 137 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 103 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 101 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 43 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41;
238 = 5 + 233 = 2 + 7 + 229 = 3 + 5 + 7 + 223 = 2 + 5 + 7 + 13 + 211 = 3 + 5 + 7 + 11 + 13 + 199 = 2 + 3 + 5 + 7 + 11 + 13 + 197 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 163 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 29 + 151 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 109 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 31 + 97 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 =  2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41;
285 = 2 + 283 = 3 + 5 + 277 = 2 + 5 + 7 + 271 = 3 + 5 + 7 + 13 + 257 = 2 + 3 + 5 + 7 + 11 + 257 = 3 + 5 + 7 + 11 + 13 + 17 + 229 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 227 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 181 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 179 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 127 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 43 + 113 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 47 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47;
338 = 7 + 331 = 2 + 5 + 331 = 3 + 5 + 13 + 317 = 2 + 3 + 5 + 11 + 317 = 3 + 5 + 7 + 13 + 17 + 293 = 2 + 3 + 5 + 7 + 11 + 17 + 293 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 263 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 257 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 211 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 41 + 197 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 139 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 137 =  3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 59 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 53;

For any n, a lower bond is given by A007504.

Q Can you verify and extend the sequence?

 

From Aug 16-22, 2025, contributions came from Ian Ianakiev, Emmanuel Vantiehem, Giorgos Kalogeropoulos, JM Rebert, Vicente Felipe Izquierdo, Gennady Gusev, Oscar Volpatti, Michael Hürter. 

***

Ian wrote:

The forth term of your sequence should be 28, since
28 = 5 + 23 = 2 + 7 + 19 = 3 + 5 + 7 + 13 = 2 + 3 + 5 + 7 + 11
Can't comment on the terms greater than 45, my code is too naive and my computer is too slow.

 ***

After receiving the Ian's email I forwarded it to Paolo, and this was his answer:

...here below the revised list. 34 and 195 were not correct: 28 and 166 are the correct ones. Now, I added also 399 (from 2 to 16 different primes).


28 = 5 + 23 = 2 + 7 + 19 = 3 + 5 + 7 + 13 = 2 + 3 + 5 + 7 + 11;
...

166 = 3 + 163 = 2 + 7 + 157 = 3 + 5 + 7 + 151 = 2 + 3 + 5 + 7 + 149 = 3 + 5 + 7 + 11 + 13 + 127 = 2 + 3 + 5 + 7 + 11 + 29 + 109 = 3 + 5 + 7 + 11 + 13 + 17 + 31 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 89 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37;
...
399 = 2 + 397 = 3 + 7 + 389 = 2 + 3 + 5 + 389 = 3 + 5 + 7 + 11 + 373 = 2 + 3 + 5 + 7 + 23 + 359 = 3 + 5 + 7 + 11 + 13 + 23 + 337 = 2 + 3 + 5 + 7 + 11 + 13 + 41 + 317 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 31 + 293 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 293 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 241 =  2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 239 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 163 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 53 + 149 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 71;

 

***

Emmanuel wrote:

It is not difficult to make a list (but it was difficult for me to find a decent program to do that)..
Here are the first  500 :
 5, 10, 21, 28, 45, 58, 81, 106, 129, 166, 201, 238, 285, 338, 399, 440, 511, 572, 645, 718, 811, 892, 993, 1064, 1173, 1268, 1401, 1484, 1615, 1730, 1869, 1988, 2139, 2276, 2439, 2594, 2769, 2924, 3111, 3266, 3459, 3638, 3835, 4028, 4255, 4454, 4665, 4888, 5121, 5356, 5625, 5840, 6115, 6350, 6639, 6870, 7153, 7422, 7729, 7992, 8293, 8586, 8931, 9210, 9541, 9870, 10225, 10538, 10891, 11258, 11623, 11978, 12349, 12734, 13123, 13502, 13909, 14306, 14715, 15116, 15571, 15968, 16429, 16856, 17301, 17744, 18193,18650, 19141, 19620, 20125, 20566, 21069, 21546, 22081, 22560, 23101, 23610, 24199, 24702, 25249, 25806, 26419, 26946, 27553, 28116, 28725, 29296, 29949, 30514, 31153, 31734, 32365, 32994, 33625, 34272, 35025, 35574, 36265, 36900, 37609, 38270, 38961, 39638, 40363, 41046, 41773, 42480, 43293, 43950, 44743, 45456, 46201, 46976, 47779, 48530, 49333, 50090, 50911, 51708, 52519, 53346, 54169, 55008, 55903, 56694, 57573, 58410, 59335, 60150, 61059, 61914, 62853, 63726, 64669, 65560, 66501, 67410, 68391, 69312, 70299, 71224, 72225, 73168, 74169, 75156, 76149, 77146, 78159, 79168, 80241, 81220, 82269, 83302, 84391, 85402, 86493, 87522, 88645, 89676, 90789, 91860, 93049, 94068, 95191, 96294, 97501, 98556, 99735, 100848, 102073, 103188, 104383, 105552, 106783, 107954, 109171, 110374, 111613, 112816, 114075, 115324, 116581, 117810, 119101, 120352, 121639, 122920, 124249, 125512, 126829, 128116, 129493, 130738, 132105, 133438, 134839, 136126, 137569, 138894, 140319, 141694, 143115, 144526, 145965, 147406, 148869, 150278, 151731, 153184, 154671, 156118, 157629, 159076, 160581, 162052, 163569, 165070, 166599, 168104, 169641, 171164, 172719, 174272, 175861, 177404, 179023, 180552, 182131, 183716, 185325, 186914, 188535, 190142, 191803, 193382, 195049, 196670, 198379, 199964, 201663, 203320, 205033, 206686, 208395, 210106, 211879, 213548, 215331, 217024, 218799, 220546, 222369, 224058, 225841, 227618, 229489, 231250, 233085, 234878, 236709, 238544, 240423, 242252, 244131, 245996, 247903, 249758, 251679, 253568, 255469, 257376, 259381, 261238, 263211, 265144, 267141, 269092, 271099, 273058, 275061, 277054, 279121, 281078, 283119, 285108, 287161, 289216, 291273, 293308, 295431, 297442, 299529, 301612, 303751, 305812, 307941, 310040, 312201, 314290, 316425, 318562, 320715, 322926, 325095, 327256, 329473, 331638, 333859, 336058, 338323, 340522, 342843, 345028, 347299, 349552, 351861, 354114, 356451, 358698, 360991, 363288, 365629, 367938, 370375, 372618, 375019, 377334, 379705, 382048, 384439, 386796, 389233, 391588, 394041, 396396, 398865, 401212, 403689, 406070, 408535, 410954, 413413, 415924, 418449, 420882, 423405, 425864, 428403, 430916, 433455, 435998, 438591, 441140, 443751, 446276, 448885, 451514, 454143, 456752, 459385, 462006, 464665, 467316, 470023, 472658, 475333, 478010, 480709, 483386, 486093, 488786, 491485, 494192, 496921, 499634, 502411, 505094, 507885, 510600, 513369, 516116, 518895, 521660, 524499, 527252, 530089, 532868, 535711, 538540, 541393, 544202, 547099, 549920, 552793, 555656, 558535, 561436, 564361, 567258, 570235, 573112, 576133, 578980, 581943, 584890, 587889, 590852, 593871, 596834, 599871, 602902, 605979, 608938, 611979, 615022, 618163, 621148, 624273, 627330, 630483, 633502, 636721, 639748, 642939, 646036, 649263, 652380, 655599, 658762, 661989, 665160, 668473, 671590, 674815, 678028, 681363, 684514, 687811, 691038, 694389, 697596, 700909, 704220, 707545, 710846, 714165, 717488, 720831, 724178, 727579, 730868, 734269, 737600, 740991, 744362, 747829, 751208, 754653, 758034, 761473, 764916, 768373, 771840, 775351, 778794, 782299, 785780, 789313, 792800, 796339, 799850, 803391, 806918, 810475, 814016, 817563, 821134, 824703, 828274.
As you can see, the fourth (28) and the tenth (166) are different from Paolo's list.  I found :
28 = 5 + 23 = 2 + 3 + 23 = 3 + 5 + 7 + 13 = 2 + 3 + 5 + 7 + 11
166 = 3 + 163 = 2 + 7 + 157 = 3 + 5 + 7 + 151 = 2 + 3 + 5 + 7 + 149 = 3 + 5 + 7 + 11 + 13 + 127 = 2 + 3 + 5 + 7 + 11 + 29 + 109 = 3 + 5 + 7 + 11 + 13 + 17 + 31 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 89 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37

***

Giorgos wrote:

Here are the first 200 terms (corrections in bold)
5,10,21,28,45,58,81,106,129,166,201,238,285,338,399,440,511,572,645,718,811,888,985,1064,1173,1268,1383,1484,
1611,1730,1869,1988,2139,2276,2439,2594,2769,2924,3111,3266,3459,3638,3835,4028,4245,4454,4665,4888,5121,5356,
5625,5840,6103,6350,6639,6870,7153,7422,7719,7992,8293,8586,8925,9210,9541,9870,10225,10538,10891,11250,11623
11978,12349,12734,13123,13502,13905,14306,14715,15116,15571,15968,16419,16850,17301,17744,18193,18650,19141,
19596,20091,20566,21069,21546,22069,22560,23101,23610,24171,24702,25249,25806,26389,26946,27553,28116,28725,
29296,29923,30514,31141,31734,32365,32994,33625,34272,34941,35574,36253,36900,37591,38260,38961,39636,40359,
41046,41773,42480,43239,43950,44713,45456,46201,46968,47779,48518,49333,50090,50911,51708,52519,53346,54169,
55008,55873,56694,57573,58410,59335,60150,61053,61914,62821,63726,64663,65560,66501,67410,68391,69312,70291,
71224,72223,73168,74163,75148,76149,77146,78159,79168,80209,81220,82269,83302,84379,85402,86493,87522,88609,
89676,90789,91860,92989,94068,95191,96294,97431,98556,99735,100848,102061,103188,104383,105552,106783,107954,
109171,110374,111601...

***

JM Reber wrote:

For k  = 11 I found :
166  = 3 + 163 = 2 + 7 + 157 = 3 + 5 + 7 + 151 = 2 + 3 + 5 + 7 + 149 = 3 + 5 + 7 + 11 + 13 + 127 = 2 + 3 + 5 + 7 + 11 + 29 + 109 = 3 + 5 + 7 + 11 + 13 + 17 + 31 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 89 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37;

***

Vicente wrote:

This is my list, it differs in 2 values and contributes 2 new ones:
5,10,21,28(differ),45,58,81,106,129,166(differ),201,238,285,338,399(new),440(new)
 
  5,{2,3}
 10,{3,7},{2,3,5}
 21,{2,19},{3,5,13},{2,3,5,11}
 28,{5,23},{2,3,23},{3,5,7,13},{2,3,5,7,11}
 45,{2,43},{3,5,37},{2,3,11,29},{3,5,7,11,19},{2,3,5,7,11,17}
 58,{5,53},{2,3,53},{3,5,7,43},{2,3,5,7,41},{3,5,7,11,13,19},{2,3,5,7,11,13,17}
 81,{2,79},{3,5,73},{2,3,5,71},{3,5,7,13,53},{2,3,5,7,11,53},{3,5,7,11,13,19,23},{2,3,5,7,11,13,17,23}
106,{3,103},{2,3,101},{3,5,19,79},{2,3,5,7,89},{3,5,7,11,13,67},{2,3,5,7,11,17,61},{3,5,7,11,13,17,19,31},
{2,3,5,7,11,13,17,19,29}
129,{2,127},{3,13,113},{2,3,11,113},{3,5,7,11,103},{2,3,5,7,11,101},{3,5,7,11,13,17,73},{2,3,5,7,11,13,17,71}
,{3,5,7,11,13,17,19,23,31},{2,3,5,7,11,13,17,19,23,29}
166,{3,163},{2,7,157},{3,5,7,151},{2,3,5,7,149},{3,5,7,11,13,127},{2,3,5,7,11,29,109},{3,5,7,11,13,17,31,79},
{2,3,5,7,11,13,17,19,89},{3,5,7,11,13,17,19,23,31,37},{2,3,5,7,11,13,17,19,23,29,37}
201,{2,199},{3,5,193},{2,3,5,191},{3,5,7,13,173},{2,3,5,7,11,173},{3,5,7,11,13,23,139},{2,3,5,7,11,13,23,137},
{3,5,7,11,13,17,19,23,103},{2,3,5,7,11,13,17,19,23,101},{3,5,7,11,13,17,19,23,29,31,43},{2,3,5,7,11,13,17,19,23,29,31,41}
238,{5,233},{2,3,233},{3,5,7,223},{2,3,5,17,211},{3,5,7,11,13,199},{2,3,5,7,11,13,197},{3,5,7,11,13,17,19,163},
{2,3,5,7,11,13,17,23,157},{3,5,7,11,13,17,19,23,31,109},{2,3,5,7,11,13,17,19,23,29,109},
{3,5,7,11,13,17,19,23,29,31,37,43},{2,3,5,7,11,13,17,19,23,29,31,37,41}
285,{2,283},{3,5,277},{2,3,11,269},{3,5,7,13,257},{2,3,5,7,11,257},{3,5,7,11,13,17,229},
{2,3,5,7,11,13,17,227},{3,5,7,11,13,17,19,29,181},{2,3,5,7,11,13,17,19,29,179},{3,5,7,11,13,17,19,23,29,31,127},
{2,3,5,7,11,13,17,19,23,29,43,113},{3,5,7,11,13,17,19,23,29,31,37,43,47},{2,3,5,7,11,13,17,19,23,29,31,37,41,47}
338,{7,331},{2,5,331},{3,5,13,317},{2,3,5,11,317},{3,5,7,11,19,293},{2,3,5,7,11,17,293},{3,5,7,11,13,17,19,263},
{2,3,5,7,11,13,17,23,257},{3,5,7,11,13,17,19,23,29,211},{2,3,5,7,11,13,17,19,23,41,197},
{3,5,7,11,13,17,19,23,29,31,41,139},{2,3,5,7,11,13,17,19,23,29,31,41,137},
{3,5,7,11,13,17,19,23,29,31,37,41,43,59},{2,3,5,7,11,13,17,19,23,29,31,37,41,47,53}
399,{2,397},{3,7,389},{2,3,5,389},{3,5,7,11,373},{2,3,5,7,23,359},{3,5,7,11,13,23,337},
{2,3,5,7,11,13,41,317},{3,5,7,11,13,17,19,31,293},{2,3,5,7,11,13,17,19,29,293},{3,5,7,11,13,17,19,23,29,31,241},
{2,3,5,7,11,13,17,19,23,29,31,239},{3,5,7,11,13,17,19,23,29,31,37,41,163},{2,3,5,7,11,13,17,19,23,29,31,37,53,149},
{3,5,7,11,13,17,19,23,29,31,37,41,43,47,73},{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,71}
440,{7,433},{2,5,433},{3,5,11,421},{2,3,5,11,419},{3,5,7,11,13,401},{2,3,5,7,11,23,389},{3,5,7,11,13,17,31,353},
{2,3,5,7,11,13,17,23,359},{3,5,7,11,13,17,19,23,29,313},{2,3,5,7,11,13,17,19,23,29,311},
{3,5,7,11,13,17,19,23,29,31,41,241},{2,3,5,7,11,13,17,19,23,29,31,41,239},{3,5,7,11,13,17,19,23,29,31,37,41,47,157},
{2,3,5,7,11,13,17,19,23,29,31,37,41,53,149},{3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,61},
{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59}

***

Gennady wrote:

2 improvements:
for n=5
28 instead of 34
28=5+23
28=2+3+23
28=3+5+7+13
28=2+3+5+7+11
 
and for n=11
166 instead 195
166=3+163
166=2+7+157
166=3+5+7+151
166=2+3+5+7+149
166=3+5+7+11+13+127
166=2+3+5+7+11+29+109
166=3+5+7+11+13+17+31+79
166=2+3+5+7+11+13+17+19+89
166=3+5+7+11+13+17+19+23+31+37
166=2+3+5+7+11+13+17+19+23+29+37
 
And extension from 16 to 30:
n=16
399=2+397
399=3+7+389
399=2+3+5+389
399=3+5+7+11+373
399=2+3+5+7+23+359
399=3+5+7+11+13+23+337
399=2+3+5+7+11+13+41+317
399=3+5+7+11+13+17+19+31+293
399=2+3+5+7+11+13+17+19+29+293
399=3+5+7+11+13+17+19+23+29+31+241
399=2+3+5+7+11+13+17+19+23+29+31+239
399=3+5+7+11+13+17+19+23+29+31+37+41+163
399=2+3+5+7+11+13+17+19+23+29+31+37+53+149
399=3+5+7+11+13+17+19+23+29+31+37+41+43+47+73
399=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+71
 
n=17
440=7+433
440=2+5+433
440=3+5+11+421
440=2+3+5+11+419
440=3+5+7+11+13+401
440=2+3+5+7+11+23+389
440=3+5+7+11+13+17+31+353
440=2+3+5+7+11+13+17+23+359
440=3+5+7+11+13+17+19+23+29+313
440=2+3+5+7+11+13+17+19+23+29+311
440=3+5+7+11+13+17+19+23+29+31+41+241
440=2+3+5+7+11+13+17+19+23+29+31+41+239
440=3+5+7+11+13+17+19+23+29+31+37+41+47+157
440=2+3+5+7+11+13+17+19+23+29+31+37+41+53+149
440=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+61
440=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59
 
n=18
511=2+509
511=3+5+503
511=2+3+7+499
511=3+5+7+17+479
511=2+3+5+7+31+463
511=3+5+7+11+13+23+449
511=2+3+5+7+11+13+31+439
511=3+5+7+11+13+17+19+47+389
511=2+3+5+7+11+13+17+19+37+397
511=3+5+7+11+13+17+19+23+29+31+353
511=2+3+5+7+11+13+17+19+23+29+71+311
511=3+5+7+11+13+17+19+23+29+31+37+47+269
511=2+3+5+7+11+13+17+19+23+29+31+37+43+271
511=3+5+7+11+13+17+19+23+29+31+37+41+43+53+179
511=2+3+5+7+11+13+17+19+23+29+31+37+41+43+67+163
511=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+73
511=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+71
 
n=19
572=31+541
572=2+13+557
572=3+5+7+557
572=2+3+5+41+521
572=3+5+7+11+23+523
572=2+3+5+7+11+23+521
572=3+5+7+11+13+17+29+487
572=2+3+5+7+11+13+17+23+491
572=3+5+7+11+13+17+19+23+31+443
572=2+3+5+7+11+13+17+19+23+29+443
572=3+5+7+11+13+17+19+23+29+31+41+373
572=2+3+5+7+11+13+17+19+23+29+31+53+53+359
572=3+5+7+11+13+17+19+23+29+31+37+41+43+293
572=2+3+5+7+11+13+17+19+23+29+31+37+41+53+281
572=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+193
572=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+191
572=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+73
572=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71
 
n=20
645=2+643
645=3+11+631
645=2+3+23+617
645=3+5+7+11+619
645=2+3+5+7+11+617
645=3+5+7+11+13+19+587
645=2+3+5+7+11+13+17+587
645=3+5+7+11+13+17+19+23+547
645=2+3+5+7+11+13+17+19+47+521
645=3+5+7+11+13+17+19+23+29+31+487
645=2+3+5+7+11+13+17+19+23+29+37+479
645=3+5+7+11+13+17+19+23+29+31+37+41+409
645=2+3+5+7+11+13+17+19+23+29+31+37+47+401
645=3+5+7+11+13+17+19+23+29+31+37+41+43+53+313
645=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+317
645=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+67+199
645=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+67+197
645=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+79
645=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+73
 
n=21
718=17+701
718=2+7+709
718=3+5+19+691
718=2+3+5+7+701
718=3+5+7+11+19+673
718=2+3+5+7+11+13+677
718=3+5+7+11+13+17+19+643
718=2+3+5+7+11+13+17+19+641
718=3+5+7+11+13+17+19+23+43+577
718=2+3+5+7+11+13+17+19+23+31+587
718=3+5+7+11+13+17+19+23+29+31+37+523
718=2+3+5+7+11+13+17+19+23+29+31+37+521
718=3+5+7+11+13+17+19+23+29+31+37+41+43+439
718=2+3+5+7+11+13+17+19+23+29+31+37+41+47+433
718=3+5+7+11+13+17+19+23+29+31+37+41+43+47+61+331
718=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+337
718=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+83+197
718=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+67+211
718=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+73+79
718=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+79
 
n=22
811=2+809
811=3+11+797
811=2+3+19+787
811=3+5+7+23+773
811=2+3+5+7+37+757
811=3+5+7+11+13+29+743
811=2+3+5+7+11+13+19+751
811=3+5+7+11+13+17+19+53+683
811=2+3+5+7+11+13+17+19+43+691
811=3+5+7+11+13+17+19+23+29+31+653
811=2+3+5+7+11+13+17+19+23+29+41+641
811=3+5+7+11+13+17+19+23+29+31+37+47+569
811=2+3+5+7+11+13+17+19+23+29+31+37+43+571
811=3+5+7+11+13+17+19+23+29+31+37+41+43+53+479
811=2+3+5+7+11+13+17+19+23+29+31+37+41+43+67+463
811=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+373
811=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+71+359
811=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+241
811=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+239
811=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+101
811=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+83+89
 
n=23
888=5+883
888=2+3+883
888=3+5+17+863
888=2+3+5+19+859
888=3+5+7+11+23+839
888=2+3+5+7+11+31+829
888=3+5+7+11+13+17+23+809
888=2+3+5+7+11+13+17+19+811
888=3+5+7+11+13+17+19+23+29+761
888=2+3+5+7+11+13+17+19+23+31+757
888=3+5+7+11+13+17+19+23+29+31+47+683
888=2+3+5+7+11+13+17+19+23+29+31+37+691
888=3+5+7+11+13+17+19+23+29+31+37+41+53+599
888=2+3+5+7+11+13+17+19+23+29+31+37+41+43+607
888=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+509
888=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+61+499
888=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+389
888=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+89+359
888=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+251
888=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+79+241
888=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+79+83+89
888=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+97
 
n=24
985=2+983
985=3+5+977
985=2+3+13+967
985=3+5+7+17+953
985=2+3+5+7+31+937
985=3+5+7+11+13+17+929
985=2+3+5+7+11+13+37+907
985=3+5+7+11+13+17+19+23+887
985=2+3+5+7+11+13+17+19+31+877
985=3+5+7+11+13+17+19+23+29+31+827
985=2+3+5+7+11+13+17+19+23+29+47+809
985=3+5+7+11+13+17+19+23+29+31+37+47+743
985=2+3+5+7+11+13+17+19+23+29+31+37+61+727
985=3+5+7+11+13+17+19+23+29+31+37+41+43+47+659
985=2+3+5+7+11+13+17+19+23+29+31+37+41+43+61+643
985=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+547
985=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+83+521
985=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+419
985=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+83+401
985=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+79+269
985=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+83+263
985=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+113
985=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+83+89+101
 
n=25
1064=3+1061
1064=2+11+1051
1064=3+5+7+1049
1064=2+3+5+23+1031
1064=3+5+7+11+17+1021
1064=2+3+5+7+11+17+1019
1064=3+5+7+11+13+17+31+977
1064=2+3+5+7+11+13+17+23+983
1064=3+5+7+11+13+17+19+23+29+937
1064=2+3+5+7+11+13+17+19+23+53+911
1064=3+5+7+11+13+17+19+23+29+31+43+863
1064=2+3+5+7+11+13+17+19+23+29+31+41+863
1064=3+5+7+11+13+17+19+23+29+31+37+41+59+769
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+53+773
1064=3+5+7+11+13+17+19+23+29+31+37+41+43+47+61+677
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+683
1064=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+79+547
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+563
1064=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+79+419
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+107+389
1064=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+83+271
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+83+269
1064=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+103
1064=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+101
 
n=26
1173=2+1171
1173=3+79+1091
1173=2++3+107+1061
1173=3++5++7+61+1097
1173=2++3++5++7+53+1103
1173=3++5++7+11+13+31+1103
1173=2++3++5++7+11+13+23+1109
1173=3++5++7+11+13+17+19+29+1069
1173=2++3++5++7+11+13+17+19+47+1049
1173=3++5++7+11+13+17+19+23+29+37+1009
1173=2++3++5++7+11+13+17+19+23+29+31+1013
1173=3++5++7+11+13+17+19+23+29+31+37+41+937
1173=2++3++5++7+11+13+17+19+23+29+31+37+47+929
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+67+827
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+53+839
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+61+733
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+733
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+607
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+601
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+463
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+461
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+101+283
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+89+293
1173=3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+103+109
1173=2++3++5++7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+113
 
n=27
1268=67+1201
1268=2+37+1229
1268=3+5+23+1237
1268=2+3+5+29+1229
1268=3+5+7+11+13+1229
1268=2+3+5+7+11+17+1223
1268=3+5+7+11+13+17+19+1193
1268=2+3+5+7+11+13+17+23+1187
1268=3+5+7+11+13+17+19+23+41+1129
1268=3+5+7+11+13+17+19+23+41+1129
1268=2+3+5+7+11+13+17+19+23+59+1109
1268=3+5+7+11+13+17+19+23+29+31+41+1069
1268=2+3+5+7+11+13+17+19+23+29+31+47+1061
1268=3+5+7+11+13+17+19+23+29+31+37+41+61+971
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+47+983
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+59+883
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+887
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+769
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+67+761
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+631
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+83+617
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+479
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+89+467
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+307
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+101+293
1268=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+109
1268=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+107
 
n=28
1383=2+1381
1383=3+7+1373
1383=2+3+5+1373
1383=3+5+7+41+1327
1383=2+3+5+7+47+1319
1383=3+5+7+11+13+17+1327
1383=5+7+11+13+17+23+1307
1383=2+3+5+7+11+13+23+1319
1383=3+5+7+11+13+17+19+29+1279
1383=2+3+5+7+11+13+17+19+23+1283
1383=3+5+7+11+13+17+19+23+29+43+1213
1383=2+3+5+7+11+13+17+19+23+29+31+1223
1383=3+5+7+11+13+17+19+23+29+31+37+59+1129
1383=2+3+5+7+11+13+17+19+23+29+31+37+83+1103
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+53+1051
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+53+1049
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+67+937
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+61+941
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+73+811
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+811
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+673
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+83+661
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+103+491
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+509
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+109+313
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+103+317
1383=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+103+109+113
1383=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+109+113
 
n=29
1484=3+1481
1484=2+11+1471
1484=3+5+17+1459
1484=2+3+5+23+1451
1484=3+5+7+11+19+1439
1484=2+3+5+7+11+17+1439
1484=3+5+7+11+13+17+19+1409
1484=5+7+11+13+17+19+31+1381
1484=2+3+5+7+11+13+17+53+1373
1484=3+5+7+11+13+17+19+23+59+1327
1484=2+3+5+7+11+13+17+19+23+83+1301
1484=3+5+7+11+13+17+19+23+29+31+37+1289
1484=2+3+5+7+11+13+17+19+23+29+31+41+1283
1484=3+5+7+11+13+17+19+23+29+31+37+41+47+1201
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+53+1193
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+61+1097
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+1103
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+79+967
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+983
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+79+839
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+89+827
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+83+691
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+89+683
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+523
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+521
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+109+317
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+107+317
1484=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+109+113
1484=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+107+113
 
n=30
1611=2+1609
1611=3+7+1601
1611=2+3+5+1601
1611=3+5+7+13+1583
1611=2+3+5+7+11+1583
1611=3+5+7+11+13+19+1553
1611=2+3+5+7+11+13+17+1553
1611=3+5+7+11+13+17+19+37+1499
1611=2+3+5+7+11+13+17+19+23+1511
1611=3+5+7+11+13+17+19+23+29+31+1453
1611=2+3+5+7+11+13+17+19+23+29+31+1451
1611=3+5+7+11+13+17+19+23+29+31+37+43+1373
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+1373
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+53+1279
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+1283
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+61+1171
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+1171
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+73+1039
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+71+1039
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+97+877
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+89+883
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+739
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+101+719
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+103+547
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+101+547
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+349
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+347
1611=3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+109+113+127
1611=2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101+103+107+109+131

***

Oscar wrote:

Paolo Lava's sequence is correct, with one exception.
For n=11 (odd), the smallest sum is s=166 (even).
After 338, next ten elements of the sequence are:
399, 440, 511, 572, 645, 718, 811, 888, 985, 1064.
in the table below are the solutions up to n=200, s=111601.
For compactness, I'm not explicitly sending prime sums too.

 

n, s n, s n, s n, s 
2  5
3  10
4  21
5  28
6  45
7  58
8  81
9  106
10  129
11  166
12  201
13  238
14  285
15  338
16  399
17  440
18  511
19  572
20  645
21  718
22  811
23  888
24  985
25  1064
26  1173
27  1268
28  1383
29  1484
30  1611
31  1730
32  1869
33  1988
34  2139
35  2276
36  2439
37  2594
38  2769
39  2924
40  3111
41  3266
42  3459
43  3638
44  3835
45  4028
46  4245
47  4454
48  4665
49  4888
50  5121
 
51  5356
52  5625
53  5840
54  6103
55  6350
56  6639
57  6870
58  7153
59  7422
60  7719
61  7992
62  8293
63  8586
64  8925
65  9210
66  9541
67  9870
68  10225
69  10538
70  10891
71  11250
72  11623
73  11978
74  12349
75  12734
76  13123
77  13502
78  13905
79  14306
80  14715
81  15116
82  15571
83  15968
84  16419
85  16850
86  17301
87  17744
88  18193
89  18650
90  19141
91  19596
92  20091
93  20566
94  21069
95  21546
96  22069
97  22560
98  23101
99  23610
100  24171
 
101  24702
102  25249
103  25806
104  26389
105  26946
106  27553
107  28116
108  28725
109  29296
110  29923
111  30514
112  31141
113  31734
114  32365
115  32994
116  33625
117  34272
118  34941
119  35574
120  36253
121  36900
122  37591
123  38260
124  38961
125  39636
126  40359
127  41046
128  41773
129  42480
130  43239
131  43950
132  44713
133  45456
134  46201
135  46968
136  47779
137  48518
138  49333
139  50090
140  50911
141  51708
142  52519
143  53346
144  54169
145  55008
146  55873
147  56694
148  57573
149  58410
150  59335
 
151  60150
152  61053
153  61914
154  62821
155  63726
156  64663
157  65560
158  66501
159  67410
160  68391
161  69312
162  70291
163  71224
164  72223
165  73168
166  74163
167  75148
168  76149
169  77146
170  78159
171  79168
172  80209
173  81220
174  82269
175  83302
176  84379
177  85402
178  86493
179  87522
180  88609
181  89676
182  90789
183  91860
184  92989
185  94068
186  95191
187  96294
188  97431
189  98556
190  99735
191  100848
192  102061
193  103188
194  104383
195  105552
196  106783
197  107954
198  109171
199  110374
200  111601

***

Michael wrote:

I found the following solutions:
 
5 = 2 + 3;
10 = 3 + 7 = 2 + 3 + 5;
21 = 2 + 19 = 3 + 5 + 13 = 2 + 3 + 5 + 11;
28 = 5 + 23 = 2 + 3 + 23 = 3 + 5 + 7 + 13 = 2 + 3 + 5 + 7 + 11;
45 = 2 + 43 = 3 + 5 + 37 = 2 + 3 + 11 + 29 = 3 + 5 + 7 + 11 + 19 = 2 + 3 + 5 + 7 + 11 + 17;
58 = 5 + 53 = 2 + 3 + 53 = 3 + 5 + 7 + 43 = 2 + 3 + 5 + 7 + 41 = 3 + 5 + 7 + 11 + 13 + 19 = 2 + 3 + 5 + 7 + 11 + 13 + 17;
81 = 2 + 79 = 3 + 5 + 73 = 2 + 3 + 5 + 71 = 3 + 5 + 7 + 13 + 53 = 2 + 3 + 5 + 7 + 11 + 53 = 3 + 5 + 7 + 11 + 13 + 19 + 23 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23;
106 = 3 + 103 = 2 + 3 + 101 = 3 + 5 + 19 + 79 = 2 + 3 + 5 + 7 + 89 = 3 + 5 + 7 + 11 + 13 + 67 = 2 + 3 + 5 + 7 + 11 + 17 + 61 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 31 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29;
129 = 2 + 127 = 3 + 13 + 113 = 2 + 3 + 11 + 113 = 3 + 5 + 7 + 11 + 103 = 2 + 3 + 5 + 7 + 11 + 101 = 3 + 5 + 7 + 11 + 13 + 17 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 71 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29;
166 = 3 + 163 = 2 + 7 + 157 = 3 + 5 + 7 + 151 = 2 + 3 + 5 + 7 + 149 = 3 + 5 + 7 + 11 + 13 + 127 = 2 + 3 + 5 + 7 + 11 + 29 + 109 = 3 + 5 + 7 + 11 + 13 + 17 + 31 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 89 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37;
201 = 2 + 199 = 3 + 5 + 193 = 2 + 3 + 5 + 191 = 3 + 5 + 7 + 13 + 173 = 2 + 3 + 5 + 7 + 11 + 173 = 3 + 5 + 7 + 11 + 13 + 23 + 139 = 2 + 3 + 5 + 7 + 11 + 13 + 23 + 137 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 103 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 101 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 43 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41;
238 = 5 + 233 = 2 + 3 + 233 = 3 + 5 + 7 + 223 = 2 + 3 + 5 + 17 + 211 = 3 + 5 + 7 + 11 + 13 + 199 = 2 + 3 + 5 + 7 + 11 + 13 + 197 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 163 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 157 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 109 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 109 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41;
285 = 2 + 283 = 3 + 5 + 277 = 2 + 3 + 11 + 269 = 3 + 5 + 7 + 13 + 257 = 2 + 3 + 5 + 7 + 11 + 257 = 3 + 5 + 7 + 11 + 13 + 17 + 229 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 227 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 181 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 179 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 127 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 43 + 113 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 47 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47;
338 = 7 + 331 = 2 + 5 + 331 = 3 + 5 + 13 + 317 = 2 + 3 + 5 + 11 + 317 = 3 + 5 + 7 + 11 + 19 + 293 = 2 + 3 + 5 + 7 + 11 + 17 + 293 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 263 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 257 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 211 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 41 + 197 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 139 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 137 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 59 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 53;
399 = 2 + 397 = 3 + 7 + 389 = 2 + 3 + 5 + 389 = 3 + 5 + 7 + 11 + 373 = 2 + 3 + 5 + 7 + 23 + 359 = 3 + 5 + 7 + 11 + 13 + 23 + 337 = 2 + 3 + 5 + 7 + 11 + 13 + 41 + 317 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 31 + 293 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 293 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 241 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 239 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 163 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 53 + 149 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 71;
440 = 7 + 433 = 2 + 5 + 433 = 3 + 5 + 11 + 421 = 2 + 3 + 5 + 11 + 419 = 3 + 5 + 7 + 11 + 13 + 401 = 2 + 3 + 5 + 7 + 11 + 23 + 389 = 3 + 5 + 7 + 11 + 13 + 17 + 31 + 353 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 359 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 313 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 311 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 241 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 239 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 157 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 53 + 149 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 61 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59;
511 = 2 + 509 = 3 + 5 + 503 = 2 + 3 + 7 + 499 = 3 + 5 + 7 + 17 + 479 = 2 + 3 + 5 + 7 + 31 + 463 = 3 + 5 + 7 + 11 + 13 + 23 + 449 = 2 + 3 + 5 + 7 + 11 + 13 + 31 + 439 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 47 + 389 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 37 + 397 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 353 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 71 + 311 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 47 + 269 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 271 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 179 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 67 + 163 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 71;
572 = 3 + 569 = 2 + 7 + 563 = 3 + 5 + 7 + 557 = 2 + 3 + 5 + 41 + 521 = 3 + 5 + 7 + 11 + 23 + 523 = 2 + 3 + 5 + 7 + 11 + 23 + 521 = 3 + 5 + 7 + 11 + 13 + 17 + 29 + 487 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 491 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 443 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 443 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 373 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 53 + 359 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 293 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 53 + 281 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 193 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 191 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 73 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71;
645 = 2 + 643 = 3 + 11 + 631 = 2 + 3 + 23 + 617 = 3 + 5 + 7 + 11 + 619 = 2 + 3 + 5 + 7 + 11 + 617 = 3 + 5 + 7 + 11 + 13 + 19 + 587 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 587 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 547 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 47 + 521 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 487 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37 + 479 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 409 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 47 + 401 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 313 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 317 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 67 + 199 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 67 + 197 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71 + 73;
718 = 17 + 701 = 2 + 7 + 709 = 3 + 5 + 19 + 691 = 2 + 3 + 5 + 7 + 701 = 3 + 5 + 7 + 11 + 19 + 673 = 2 + 3 + 5 + 7 + 11 + 13 + 677 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 643 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 641 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 43 + 577 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 587 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 523 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 521 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 439 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 433 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 61 + 331 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 337 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 83 + 197 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 67 + 211 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 73 + 79 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 79;
811 = 2 + 809 = 3 + 11 + 797 = 2 + 3 + 19 + 787 = 3 + 5 + 7 + 23 + 773 = 2 + 3 + 5 + 7 + 37 + 757 = 3 + 5 + 7 + 11 + 13 + 29 + 743 = 2 + 3 + 5 + 7 + 11 + 13 + 19 + 751 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 53 + 683 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 43 + 691 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 653 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 41 + 641 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 47 + 569 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 571 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 479 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 67 + 463 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 373 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 71 + 359 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71 + 241 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71 + 239 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 101 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 83 + 89;
888 = 5 + 883 = 2 + 3 + 883 = 3 + 5 + 17 + 863 = 2 + 3 + 5 + 19 + 859 = 3 + 5 + 7 + 11 + 23 + 839 = 2 + 3 + 5 + 7 + 11 + 31 + 829 = 3 + 5 + 7 + 11 + 13 + 17 + 23 + 809 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 811 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 761 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 757 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 47 + 683 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 691 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 53 + 599 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 607 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 509 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 61 + 499 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 389 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 89 + 359 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 251 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 79 + 241 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 79 + 83 + 89 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 97;
985 = 2 + 983 = 3 + 5 + 977 = 2 + 3 + 13 + 967 = 3 + 5 + 7 + 17 + 953 = 2 + 3 + 5 + 7 + 31 + 937 = 3 + 5 + 7 + 11 + 13 + 17 + 929 = 2 + 3 + 5 + 7 + 11 + 13 + 37 + 907 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 887 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 31 + 877 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 827 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 47 + 809 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 47 + 743 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 61 + 727 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 659 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 61 + 643 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 547 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 83 + 521 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 419 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 83 + 401 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 79 + 269 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 83 + 263 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 113 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 83 + 89 + 101;
 
For n = 5 I have a different result 28.
For n = 11 I have a different result 166.

***

On Aug 28, 2025, Oscar Volpatti wrote again:

I searched up to sums of 1000 distinct primes.
Largest solution found for puzzle 1233:
n = 1000, s = 3682963.
See attached files:
Pu1233 OV a.txt for solutions with index n <= 1000;
Pu1233 OV b.txt for the minimal sums corresponding to solution s = 3682963.
Solutions with index n  <= 501 are consistent with those previously submitted by Emmanuel Vantieghem, with 360 matches and 140 improved values.

 

***

 

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