Problems & Puzzles: Conjectures

Conjecture 93. Related to A103514

Alain Rochelli sent on April, 2022, the following conjecture:

The OEIS sequence A103514 is related to this :

"a(n) is the smallest m such that primorial(n)/2 - 2^m is prime, with n>=2"

Examples :

n=3 ; a(3)=1; p(3)#/2 - 2^1 = 5#/2 - 2^1 = 30/2 - 2 = 13  prime
n=4 ; a(4)=1; p(4)#/2 - 2^1 = 7#/2 - 2^1 = 210/2 - 2 = 103  prime
n=14 ; a(14)=25; p(14)#/2 - 2^25 = 43#/2 - 2^25 = 6 541 380 632 280 583  prime

We can observe that the average A(n) of a(n)*log(p(n))/p(n) grows approximately from 0,2 to 0,37 when p(n) varies from p(3)=5 to p(87)=449.

For n tending to infinity, we conjecture that A(n) converges to 0,56146, equal to 1/e^G where G is Euler-Mascheroni constant.

A(n) = 1/n Sum_{i=3 .. n} a(i)*log(p(i))/p(i) = 1/n Sum_{i=3 .. n} a(i)/(p(i)/log(p(i))) with p(i) as the ith prime

The growth of A(n) is very slow and exhibits fluctuations. For example, we need to go to p(900)=6997 to get A(900)=0.52...

A local average 1/(b-a) Sum_{i=a .. b} on a segment [p(a), p(b)] is too fluctuating to guess a convergence value.
G is Euler's constant equal to 0.5772 ... and e^G=exp(G)=1.7810724

Q1. Send your extension of A103514 to get a better convergence value of A(n).

Q2. Try to find a heuristic proof for this conjecture.

During the week from April 30 to May 6, contributions came from Simon Cavegn, Alain Rochelli

***

Simon wrote:

Extension of A103514, using a probable prime test:
0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 25, 2, 1, 6, 6, 19, 1, 13, 3, 3, 11, 29, 2, 1, 6, 3, 4, 2, 6, 4, 15, 6, 4, 20, 4, 1, 7, 16, 4, 7, 22, 3, 12, 13, 9, 35, 2, 3, 3, 52, 35, 3, 32, 15, 13, 10, 53, 56, 9, 16, 36, 5, 8, 5, 22, 3, 14, 2, 64, 37, 8, 22, 42, 11, 22, 22, 12, 11, 26, 1, 54, 187, 20, 9, 36, 6, 101, 48, 57, 15, 37, 5, 10, 140, 3, 16, 5, 31, 32, 1, 68, 18, 141, 114, 5, 14, 133, 44, 9, 50, 4, 6, 44, 6, 88, 55, 20, 39, 1, 48, 21, 4, 4, 41, 35, 1, 61, 9, 48, 23, 33, 26, 195, 29, 37, 76, 32, 43, 29, 76, 60, 1, 28, 70, 13, 29, 30, 85, 139, 29, 381, 74, 67, 3, 257, 71, 28, 24, 51, 69, 65, 30, 135, 29, 8, 129, 170, 61, 180, 54, 222, 145, 45, 32, 95, 152, 32, 4, 31, 29, 25, 20, 2, 101, 188, 99, 15, 12, 40, 202, 170, 65, 139, 84, 367, 48, 18, 161, 14, 54, 60, 67, 440, 53, 112, 27, 141, 89, 30, 69, 110, 11, 14, 12, 300, 82, 115, 72, 26, 200, 6, 201, 137, 170, 328, 99, 55, 253, 74, 11, 306, 37, 21, 224, 192, 147, 102, 6, 112, 143, 230, 164, 202, 37, 342, 96, 54, 33, 12, 8, 342, 73, 32, 42, 32, 12, 93, 174, 91, 37, 35, 346, 166, 195, 3, 33, 239, 30, 15, 2, 30, 112, 126, 122, 193, 1, 662, 5, 71, 126, 55, 63, 117, 381, 64, 32, 16, 4, 68, 75, 461, 135, 4, 174, 169, 142, 35, 75, 41, 130, 21, 30, 50, 77, 99, 322, 52, 54, 149, 50, 41, 147, 94, 57, 10, 94, 20, 25, 354, 197, 315, 186, 15, 93, 39, 79, 37, 195, 27, 571, 100, 26, 403, 22, 149, 15, 15, 199, 332, 338, 90, 121, 109, 92, 305, 136, 243, 265, 293, 6, 133, 59, 244, 119, 40, 16, 34, 133, 306, 159, 108, 51, 2, 3, 154, 110, 298, 586, 12, 38, 98, 295, 182, 279, 450, 16, 295, 61, 314, 219, 46, 46, 4, 10, 141, 88, 365, 7, 106, 1, 529, 327, 70, 312, 68, 129, 9, 132, 103, 41, 34, 416, 266, 872, 470, 369, 282, 108, 71, 150, 14, 828, 313, 195, 279, 90, 85, 117, 97, 91, 850, 484, 136, 87, 20, 350, 26, 337, 163, 213, 77, 7, 107, 67, 149, 551, 6, 158, 92, 129, 59, 70, 363, 206, 278, 28, 192, 411, 135, 83, 19, 354, 599, 79, 8, 708, 19, 81, 37, 89, 278, 75, 592, 224, 233, 26, 63, 50, 22, 157, 264, 310, 3, 126, 516, 30, 94, 3, 130, 12, 320, 238, 850, 87, 88, 611, 217, 120, 748, 30, 981, 192, 84, 411, 243, 62, 82, 20, 78, 21, 50, 77, 163, 34, 311, 19, 236, 334, 99, 70, 196, 66, 85, 208, 390, 14, 349, 71, 207, 91, 40, 62, 259, 382, 146, 19, 1125, 50, 158, 106, 615, 93, 687, 64, 50, 665, 90, 469, 70, 311, 199, 114, 159, 68, 456, 137, 807, 18, 101, 263, 98, 216, 711, 133, 87, 1067, 187, 124, 709, 311, 254, 373, 183, 96, 49, 310, 75, 773, 837, 1255, 108, 142, 32, 21, 49, 402, 246, 1169, 114, 216, 630, 47, 26, 239, 205, 381, 172, 501, 54, 254, 119, 633, 73, 58, 211, 67, 50, 110, 505, 462, 60, 380, 430, 350, 1, 134, 734, 276, 95, 51, 57, 77, 50, 76, 172, 161, 276, 561, 597, 999, 181, 529, 151, 491, 124, 126, 195, 30, 126, 69, 931, 298, 233, 73, 352, 59, 112, 79, 67, 250, 783, 198, 99, 145, 74, 589, 77, 319, 250, 191, 91, 1880, 177, 104, 342, 232, 28, 1064, 1263, 936, 30, 147, 291, 295, 428, 132, 401, 89, 143, 193, 859, 48, 183, 807, 15, 20, 162, 207, 738, 216, 330, 44, 84, 851, 898, 17, 242, 195, 430, 360, 301, 95, 209, 266, 660, 1619, 22, 1956, 300, 121, 243, 473, 1117, 816, 1238, 189, 673, 203, 195, 447, 513, 82, 359, 55, 344, 681, 273, 309, 166, 83, 821, 272, 351, 44, 449, 283, 127, 166, 76, 83, 307, 407, 120, 114, 51, 258, 19, 120, 549, 27, 495, 447, 778, 127, 1127, 173, 307, 1010, 192, 63, 481, 231, 674, 160, 243, 737, 169, 171, 75, 357, 319, 107, 1048, 708, 205, 498, 288, 60, 13, 122, 33, 819, 144, 169, 389, 282, 439, 513, 633, 75, 714, 445, 1325, 572, 468, 68, 179, 324, 794, 116, 482, 1221, 564, 680, 1, 202, 100, 21, 755, 732, 719, 23, 45, 397, 272, 707, 162, 175, 21, 800, 121, 533, 468, 436, 962, 605, 694, 40, 247, 784, 62, 196, 1453, 1114, 841, 128, 278, 2550, 597, 469, 649, 5, 1047, 233, 156, 535, 646, 420, 824, 251, 370, 716, 1333, 126, 445, 284, 907, 721, 236, 212, 254, 605, 134, 431, 66, 209, 1205, 148, 1442, 850, 122, 429, 1173, 824, 481, 1587, 77, 71, 127, 161, 48, 180, 1269, 218, 78, 330, 101, 326, 1010, 427, 40, 162, 212, 294, 96, 175, 396, 158, 361, 189, 5, 649, 60, 964, 55, 216, 678, 223, 76, 51, 1416, 152, 338, 1571, 226, 737, 86, 1735, 166, 433, 1029, 785, 38, 99, 273, 724, 766, 693, 483, 24, 408, 23, 426, 477, 204, 1, 566, 33, 60, 276, 33, 588, 129, 122, 10, 63, 666, 697, 323, 51, 955, 233, 79, 105, 509, 204, 267, 99, 304, 326, 1064, 482, 154, 142, 120, 2102, 75, 761, 1259, 128, 92, 21, 577, 34, 1373, 869, 748, 466, 361, 526, 289, 400, 72, 18, 10, 97, 193, 55, 236, 481, 1092, 351, 424, 466, 116, 119, 271, 249, 959, 293, 312, 207, 727, 632, 286, 472, 37, 244, 47, 36, 113, 233, 383, 658, 763, 116, 290, 857, 777, 40, 430, 490, 263, 177, 71, 74, 144, 580, 335, 199, 456, 108, 78, 98, 436, 304, 24, 1858, 1019, 45, 73, 167, 153, 36, 786, 303, 1294, 75, 463, 225, 64, 353, 38, 293, 121, 989, 841, 950, 335, 23, 317, 236, 1251, 334, 163, 1237, 98, 214, 682, 460, 545, 387, 370, 54, 388, 97, 396, 1211, 579, 489, 617, 298, 316, 333, 259, 818, 796, 201, 349, 187, 105, 906, 187, 260, 1346, 45, 903, 1590, 760, 1049, 49, 264, 332, 521, 50, 118, 343, 660, 1263, 5, 205, 322, 264, 392, 496, 154, 255, 1088, 942, 500, 36, 787, 192, 413, 51, 118, 201, 563, 1187, 18, 294, 699, 173, 612, 661, 125, 184, 1028, 275, 277, 698, 1284, 102, 23, 865, 110, 1336, 550, 319, 130, 1183, 957, 302, 547, 159, 424, 318, 238, 257, 6, 319, 1191, 1021, 122, 119, 135, 655, 822, 155, 23, 1, 704, 164, 291, 512, 1801, 732, 592, 436, 403, 1397, 115, 895, 22, 37, 775, 409, 308, 1221, 12, 179, 525, 832, 821, 158, 437, 73, 276, 348, 24, 462, 1441, 602, 52, 613, 402, 699, 216, 1277, 276, 422, 495, 1139, 247, 264, 2121, 930, 90, 241, 23, 276, 276, 210, 480, 712, 265, 1054, 1313, 2323, 777, 1181, 587, 2007, 658, 2043, 231, 37, 1032, 360, 201, 192, 460, 1171, 451, 255, 504, 158, 899, 41, 127, 1643, 551, 307, 160, 1358, 117, 1273, 160, 237, 141, 116, 41, 947, 40, 214, 69, 1179, 206, 250, 1440, 718, 166, 1, 152, 51, 485, 2990, 50, 583, 149, 662, 629, 664, 361, 1121, 123, 150, 767, 307, 470, 1426, 222, 243, 1780, 4243, 392, 57, 145, 335, 577, 204, 1482, 1698, 149, 153, 1661, 72, 158, 524, 24, 1140, 1665, 1055, 312, 758, 97, 8, 124, 669, 1189, 374, 286, 301, 283, 1061, 925, 216, 441, 880, 19, 1059, 1448, 117, 609, 72, 185, 106, 158, 93, 683, 207, 145, 665, 29, 1627, 484, 1161, 148, 759, 2373, 419, 788, 1525, 1632, 173, 553, 1195, 482, 1056, 904, 1715, 661, 216, 447, 188, 902, 641, 53, 28, 1145, 1477, 1113, 252, 685, 1762, 1856, 237, 838, 158, 250, 936, 1493, 779, 1731, 139, 279, 1149, 50, 346, 161, 438, 3747, 316, 1711, 1238, 1378, 246, 104, 429, 589, 717, 42, 124, 290, 428, 248, 160, 303, 121, 441, 315, 57, 38, 626, 1759, 241, 500, 2553, 67, 920, 593, 4, 153, 272, 59, 870, 667, 100, 672, 562, 257, 409, 74, 169, 95, 527, 25, 455, 171, 500, 1433, 573, 117, 247, 232, 1483, 1149, 1034, 1249, 1103, 457, 735, 1930, 90, 237, 692, 69, 2052, 115, 74, 115, 875, 883, 1573, 441, 361, 181, 205, 838, 856, 333, 1100, 383, 581, 179, 182, 237, 133, 337, 1045

***

Alain wrote:

Let p(n) denote the nth prime. Using the Tschebycheff function Tf[p(n)] = Sum_{i=1 .. n} log(p(i)), a way of formulating the famous prime number theorem asserts that Tf[p(n)] ~ p(n)
Also, primorial(n) = p(n)# = e^[log(p(n)#)] = e^[Tf(p(n))] ~ e^p(n) and p(n)#/2 ~ e^p(n) / 2

m integer such that p(n)#/2 – 2^m > 0 involves m < k ~ log[e^p(n)/2] / log2 = (p(n)-log2) / log2 = [p(n)/log2] - 1

For p(n) large enough (n tending to infinity), when m traverses the sequence {m=1 .. k} the average of 2^m is equal to 1/k * Sum_{m=1 .. k} 2^m = 2/k * (2^k-1) ~ 2^(k+1) / k
By replacing k+1 by p(n)/log2 we get that average of 2^m is equivalent to [2^(p(n)/log2)] / [(p(n)/log2) - 1] ~ 2^p(n) / p(n)
and 2^p(n) / p(n) is negligible compared to e^p(n) / 2, i.e. ~ p(n)#/2

Consequently, the probability that p(n)#/2 – 2^m is prime is approximately equal to 1/log(p(n)#/2) ~ 1/(p(n)-log2) - for all values of m in the sequence {m=1 .. k} - multiplied by e^G * log(p(n)) because p(n)#/2 – 2^m has no prime factor <= p(n) (using Mertens formula).

Let Kmean denote the average number of m for which p(n)#/2 – 2^m is prime. Kmean is equal to k * e^G * log(p(n)) / (p(n)-log2)
or even equal to e^G * log(p(n)) / log2 by replacing k by (p(n)-log2) / log2

Finally, the average of the smallest m such that p(n)#/2 – 2^m is prime is given by k / Kmean
or even equal to [(p(n)-log2) / log2] / [e^G * log(p(n)) / log2] which is equivalent to (1/e^G) * p(n) / log(p(n))

***

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