Problems & Puzzles: Puzzles

Puzzle 1026. Another pine-like-prime puzzle

Let's finish this incredible troubling year, 2020, with a "pine-like-triangle" prime puzzle, in order to celebrate the new and becoming year 2021. Hopefully a better year than the previous one.

The author of this puzzle is Metin Sariyar, who modified a bit a previous puzzle by G.L. Honaker, Jr. (see https://oeis.org/A047837 )

The pine-like-triangle with n rows, each row with 2i-1 primes, for i=1 to n, such that each row sums to the same prime p. No one single prime is repeated in a pine-like-triangle.

The target sequence is the smallest primes with this property. The first three p terms Sariyar found are : 2, 19, 53,...

The pine-like-triangles, for rows n=1, 2 & 3 are:

 2 19 3 5 11 53 3 19 31 5 7 11 13 17

Q. Find the smallest p for n=4 to 10?

Solutions came, from Jan 2 to 8 from Emmanuel Vantieghem, Giorgos Kalogeropoulos, Michael Hürter, Oscar Volpatti, Paul Clear, Fausto Morales.

***

Emmanuel wrote:

I found solutions for  n = 4  and  n = 5 :

n = 4 :
131
17, 53, 61
3, 7, 37, 41, 43
5, 11, 13, 19, 23, 29, 31

n = 5
269
71, 97, 101
3, 31, 67, 79, 89
5, 7, 11, 53, 59, 61, 73
13, 17, 19, 23, 29, 37, 41, 43, 47

n = 6 :
503
163, 167, 173
3, 79, 127, 137, 157
5, 7, 59, 103, 107, 109, 113
11, 13, 17, 19, 73, 83, 89, 97, 101
23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71

n = 7 :
853
277, 283, 293
43, 181, 193, 197, 239
3, 5, 139, 163, 173, 179, 191
7, 11, 13, 61, 137, 149, 151, 157, 167
17, 19, 23, 29, 83, 101, 103, 107, 113, 127, 131
31, 37, 41, 47, 53, 59, 67, 71, 73, 79, 89, 97, 109

I' think they are 'minimal'.

***

Giorgos wrote:

Firstly I tried to find a formula that gives me a lower bound for every n.
The formula
seq[t_] := NextPrime@Floor[Total[Prime@Range[Sum[2 k + 1, {k, 0, t}]]]/t]  (written in Mathematica) returned
{2, 19, 53, 131, 269, 487, 821, 1277, 1901, 2683, 3691, 4931, 6449, 8219, 10331, 12791, 15619, 18869, 22567, 26729, 31379...}

which I used as starting points for my research.
The final terms for n = 1...10  that I managed to confirm with examples are:
(I don't know if they are minimal)
{2, 19, 53, 131, 269, 503, 853, 1361, 1999, 2879}

n=4    {131}
{17,53,61}
{11,23,29,31,37}
{3,5,7,13,19,41,43}

n=5         {269}
{79,89,101}
{17,23,59,73,97}
{7,13,19,31,61,67,71}
{3,5,11,29,37,41,43,47,53}

n=6                  {503}
{163,167,173}
{79,83,101,113,127}
{17,23,29,31,109,137,157}
{3,7,11,13,73,89,97,103,107}
{5,19,37,41,43,47,53,59,61,67,71}

n=7                       {853}
{277,283,293}
{139,151,157,167,239}
{31,37,103,149,163,173,197}
{11,13,19,23,43,179,181,191,193}
{5,7,29,41,47,107,109,113,127,131,137}
{3,17,53,59,61,67,71,73,79,83,89,97,101}

n=8                              {1361}
{443,457,461}
{229,241,271,283,337}
{31,43,191,223,227,293,353}
{11,17,19,41,233,251,257,263,269}
{3,5,7,13,173,179,181,193,197,199,211}
{29,37,47,53,61,71,137,139,149,151,157,163,167}
{23,59,67,73,79,83,89,97,101,103,107,109,113,127,131}

n=9                                         {1999}
{643,673,683}
{349,373,383,431,463}
{41,281,313,317,331,337,379}
{11,13,17,263,269,347,353,359,367}
{29,37,59,61,71,271,277,283,293,307,311}
{3,5,7,31,53,223,227,229,233,239,241,251,257}
{19,23,47,67,73,79,167,173,179,181,191,193,197,199,211}
{43,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163}

n=10                                              {2879}
{941,967,971}
{457,503,631,641,647}
{263,397,409,439,443,449,479}
{29,41,53,379,433,467,487,491,499}
{7,13,19,23,373,383,389,401,419,421,431}
{3,5,31,37,43,317,331,337,347,349,353,359,367}
{11,17,47,59,61,79,269,271,277,281,283,293,307,311,313}
{67,83,89,97,103,107,109,113,211,223,227,229,233,239,241,251,257}
{71,73,101,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199}

***

Michael wrote:

n = 4:

131
17 47 67
7 13 37 43 31
5 3 41 19 11 29 23

n = 5:

269
79 89 101
29 71 73 37 59
41 7 23 67 47 31 53
43 3 5 11 17 13 19 61 97

n = 6:

503
167 139 197
31 107 157 71 137
109 47 37 41 79 89 101
3 43 61 127 53 29 103 67 17
59 13 113 83 5 23 11 97 7 73 19

n = 7:

853
211 311 331
151 107 163 193 239
191 127 139 73 179 137 7
181 17 197 61 3 97 167 101 29
67 19 173 41 149 83 113 5 71 79 53
103 43 13 89 11 157 37 59 109 31 23 47 131

n = 8:

1361
461 569 331
223 269 241 281 347
101 271 103 233 293 167 193
17 127 173 197 109 211 239 251 37
181 3 29 47 107 137 71 257 179 199 151
23 41 67 5 149 43 277 131 59 97 283 113 73
7 89 53 61 83 79 163 229 227 139 19 31 157 13 11

n = 9:

1999
509 661 829
397 373 379 431 419
137 313 269 353 337 349 241
149 89 173 227 347 233 311 331 139
389 257 199 367 157 113 151 43 23 229 71
277 41 37 239 197 163 271 79 53 97 61 167 317
281 211 7 29 181 131 283 101 17 3 11 293 73 251 127
263 103 47 67 59 307 193 83 19 191 13 107 5 223 31 109 179

n = 10:

2879
829 677 1373
587 467 673 653 499
443 283 449 431 563 331 379
383 313 293 317 463 433 269 7 401
397 137 241 199 311 223 461 229 263 359 59
211 37 29 277 113 457 421 353 97 349 271 181 83
139 149 163 151 157 103 179 479 41 191 109 251 47 347 373
233 101 257 419 19 31 239 127 71 307 389 193 11 3 173 227 79
61 281 367 131 89 17 5 23 167 439 409 13 53 73 197 107 337 67 43

They could be minimal, even for n = 10, but I am not 100% sure.

I could only compute the following bounds:

3 48.99
4 126.33
5 264.49
6 484.99
7 814.33
8 1270.14
9 1889.24
10 2681.22

***

Oscar wrote:

Dear carlos,
the first 15 terms of the pine-like-prime sequence are:
2, 19, 53, 131, 269, 503, 853, 1361, 1999, 2879, 3989, 5323?, 6983, 8971, 11299...

Related pine-like-triangles (listed only for n > 3):

131

37 41 53

17 19 23 29 43

3 5 7 11 13 31 61

269

79 89 101

43 47 53 59 67

17 23 29 31 37 61 71

3 5 7 11 13 19 41 73 97

503

163 167 173

89 97 101 107 109

53 59 61 71 73 83 103

17 29 31 37 41 43 47 127 131

3 5 7 11 13 19 23 67 79 137 139

853

277 283 293

157 163 173 179 181

103 107 109 127 131 137 139

67 73 79 83 89 97 101 113 151

23 29 37 41 43 53 59 61 149 167 191

3 5 7 11 13 17 19 31 47 71 197 199 233

1361

443 457 461

263 269 271 277 281

167 179 191 193 197 211 223

113 131 137 139 149 157 163 173 199

67 79 83 89 97 103 109 127 151 227 229

29 37 41 43 53 59 61 71 73 181 233 239 241

3 5 7 11 13 17 19 23 31 47 101 251 257 283 293

1999

643 673 683

373 397 401 409 419

269 271 277 281 283 307 311

197 199 211 223 227 229 233 239 241

137 149 151 163 167 173 179 181 191 251 257

73 97 101 103 107 109 113 131 139 157 263 293 313

31 37 43 47 59 61 67 71 79 83 89 317 331 337 347

3 5 7 11 13 17 19 23 29 41 53 127 193 353 359 367 379

2879

941 967 971

563 569 571 577 599

389 397 401 409 419 431 433

281 293 311 313 317 331 337 347 349

229 233 239 251 257 263 269 271 277 283 307

163 173 179 181 191 193 197 199 223 227 241 353 359

97 101 103 109 113 127 131 139 149 151 157 367 373 379 383

43 47 53 59 61 67 71 73 79 83 107 137 211 439 443 449 457

3 5 7 11 13 17 19 23 29 31 37 41 89 167 463 467 479 487 491

3989

1307 1321 1361

773 787 797 811 821

541 557 563 571 577 587 593

419 421 433 439 443 449 457 461 467

311 331 347 353 359 367 373 379 383 389 397

251 263 269 271 277 281 293 307 313 317 337 401 409

191 197 199 211 223 227 229 233 239 241 257 283 349 431 479

113 131 137 139 149 151 157 163 167 173 179 181 193 463 491 499 503

53 59 61 71 73 79 83 89 97 101 103 107 109 127 487 521 569 599 601

3 5 7 11 13 17 19 23 29 31 37 41 43 47 67 509 607 613 617 619 631

5323

1759 1777 1787

1049 1051 1063 1069 1091

739 743 751 761 769 773 787

557 571 577 587 593 601 607 613 617

449 457 461 467 479 487 491 499 503 509 521

359 379 383 389 397 401 409 419 421 431 433 439 463

271 283 307 311 313 317 331 337 347 349 353 367 373 523 541

211 223 227 229 233 239 241 251 257 263 269 277 281 443 547 563 569

109 127 131 137 139 149 151 163 167 173 179 181 191 193 599 619 631 641 643

53 59 61 67 71 73 79 83 89 97 101 103 107 113 197 647 653 659 661 673 677

3 5 7 11 13 17 19 23 29 31 37 41 43 47 157 199 293 683 691 701 719 757 797

6983

2311 2333 2339

1367 1381 1399 1409 1427

971 983 991 997 1009 1013 1019

733 743 761 769 773 787 797 809 811

599 613 617 619 631 641 643 647 653 659 661

487 499 503 509 521 523 541 547 557 563 569 577 587

409 419 421 431 433 439 449 457 461 463 467 479 491 571 593

311 317 331 347 349 353 359 367 373 379 383 389 401 443 601 607 673

233 239 241 251 257 263 269 271 277 283 293 307 313 337 397 677 683 691 701

131 139 149 151 157 163 167 173 179 181 191 193 197 199 211 709 719 727 739 751 757

59 67 71 73 79 83 89 97 101 107 109 113 127 137 223 227 229 821 823 827 829 839 853

3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 61 103 281 857 863 877 881 883 887 911

8971

2971 2999 3001

1783 1787 1789 1801 1811

1237 1277 1279 1289 1291 1297 1301

967 971 983 991 997 1009 1013 1019 1021

757 787 797 809 811 821 823 827 829 853 857

647 653 659 661 673 677 683 691 709 719 727 733 739

557 563 569 571 577 587 593 599 601 607 613 619 631 641 643

449 457 461 463 467 479 491 499 503 509 521 523 541 547 617 701 743

347 353 359 367 373 379 389 397 401 409 419 421 431 433 439 751 761 769 773

263 269 271 277 281 283 293 307 311 313 317 331 337 349 383 443 487 839 859 877 881

167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 863 883 887 907 911 929

29 41 43 47 53 59 61 67 71 73 83 89 97 101 103 107 109 919 937 941 947 953 977 1031 1033

3 5 7 11 13 17 19 23 31 37 79 113 127 131 137 139 149 151 157 163 1039 1049 1061 1063 1069 1087 1091

11299

3739 3767 3793

2239 2251 2267 2269 2273

1597 1601 1607 1609 1621 1627 1637

1223 1229 1231 1237 1249 1277 1279 1283 1291

983 997 1013 1019 1021 1031 1033 1039 1049 1051 1063

823 827 839 853 857 859 863 877 881 883 907 911 919

701 709 719 727 733 739 743 751 757 761 769 773 797 809 811

599 601 613 619 631 641 643 647 653 659 661 673 677 683 691 787 821

479 487 491 503 509 521 523 541 547 557 563 569 577 587 593 607 829 887 929

379 383 397 409 419 421 431 433 439 443 449 457 461 463 467 499 571 937 941 947 953

269 281 293 307 311 313 317 331 337 347 349 353 359 367 373 389 401 617 967 971 977 1009 1061

157 163 173 179 181 191 193 197 199 211 223 227 229 233 241 251 257 263 991 1069 1087 1091 1093 1097 1103

67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 151 167 239 1109 1117 1123 1129 1151 1153 1163 1171

3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 149 271 277 283 1181 1187 1193 1201 1213 1259 1289 1297

About term a(12)= 5323: lower bound a(12) >= 5309 holds; I could found no pine-like-triangle matching it, but I didn't perform an exaustive search.
Best regards.

***

Paul wrote:

Here’s a solution for n = 4

131

37 41 53

11 17 19 23 61

3 5 7 13 29 31 43

***

Fausto wrote:

Minimal solutions for n = 4 to 7:

p(4) = 131

131
29  41  61
3  17  31  37  43
5   7  11  13  19  23  53

p(5) = 269

269
71  97  101
17  29   61  73  89
5   7   11   47  53  79  67
3   13  23  19  31  37  41  43  59

p(6) = 499

499
149 157 193
23   97  107  109 163
3    47   71    83    79    89   127
29   31   37   41   43    59    61    67   131
5    7    11   13   17   19    53    73    97   101  103

p(7) = 853

853
269  271  313
73  151  233  197  199
41   53   59    167  173  179  181
3    31   61   67    79    101  191  157  163
5     7    11   13   71   103   109  127  131  137  139
17   19   23   29   37   43    47     83     89    97   107  113  149

***

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