This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A078721 #36 Apr 18 2025 21:02:45
%S A078721 2,3,7,17,31,53,79,109,157,199,263,331,401,479,577,661,773,887,1021,
%T A078721 1153,1297,1459,1609,1787,1993,2161,2377,2609,2797,3041,3313,3547,
%U A078721 3803,4079,4363,4663,4987,5309,5647,5953,6311,6689,7027,7481,7841,8263,8689
%N A078721 a(n) = prime(n*(n+1)/2 + 1).
%C A078721 Primes on the left side of the triangle formed by listing successively the prime numbers in a triangular grid:
%C A078721 2
%C A078721 3 5
%C A078721 7 11 13
%C A078721 17 19 23 29
%C A078721 31 37 41 43 47
%C A078721 53 59 61 67 71 73
%C A078721 The sum of the reciprocals appears to converge.
%C A078721 As the terms grow faster than the triangular numbers and the sum of inverse numbers converges, the sum of inverses indeed converges. - _Joerg Arndt_, Oct 28 2015
%C A078721 When arranged as a rectangular array, every row eventually intersperses every other row and every column eventually intersperses every other column; see example. - _Clark Kimberling_, Apr 13 2025
%H A078721 Michael De Vlieger, <a href="/A078721/b078721.txt">Table of n, a(n) for n = 0..10000</a>
%F A078721 a(n) = A000040(A000124(n)). - _Altug Alkan_, Oct 28 2015
%F A078721 a(n) = A151800(A011756(n)) for n >= 1. - _Amiram Eldar_, Sep 05 2024
%e A078721 2 3 7 17 31 53 79 109 157 199
%e A078721 5 11 19 37 59 83 113 163 211 269
%e A078721 13 23 41 61 89 127 167 223 271 347
%e A078721 29 43 67 97 131 173 227 277 349 421
%e A078721 47 71 101 137 179 229 281 353 431 503
%e A078721 73 103 139 181 233 283 359 433 509 607
%e A078721 107 149 191 239 293 367 439 521 613 709
%e A078721 151 193 241 307 373 443 523 617 719 827
%e A078721 197 251 311 379 449 541 619 727 829 953
%e A078721 257 313 383 457 547 631 733 839 967 1087
%e A078721 Interspersion of rows 3 and 5 begins with
%e A078721 41 61 89 127...
%e A078721 43 67 97...
%e A078721 Interspersion of columns 3 and 5 begins with
%e A078721 19
%e A078721 31
%e A078721 41
%e A078721 59
%e A078721 67
%e A078721 89
%t A078721 Table[Prime[n (n + 1)/2 + 1], {n, 0, 46}] (* _Michael De Vlieger_, Oct 28 2015 *)
%t A078721 Prime[#]&/@(Accumulate[Range[0,50]]+1) (* _Harvey P. Dale_, Aug 04 2018 *)
%t A078721 Grid[Table[Prime[n + (n + k - 2) (n + k - 1)/2], {n, 1, 20}, {k, 1, 15}]]
%t A078721 (* _Clark Kimberling_, Apr 13 2025 *)
%o A078721 (PARI) a(n) = prime(n*(n+1)/2 + 1);
%o A078721 (Magma) [NthPrime(n*(n + 1) div 2+1): n in [0..50]]; // _Vincenzo Librandi_, Jun 08 2016
%Y A078721 Cf. A000040, A011756, A078722, A078723, A078724, A078725, A151800.
%Y A078721 Cf. A114537.
%K A078721 nonn,easy
%O A078721 0,1
%A A078721 _Cino Hilliard_, Dec 20 2002