A078721 a(n) = prime(n*(n+1)/2 + 1).

2, 3, 7, 17, 31, 53, 79, 109, 157, 199, 263, 331, 401, 479, 577, 661, 773, 887, 1021, 1153, 1297, 1459, 1609, 1787, 1993, 2161, 2377, 2609, 2797, 3041, 3313, 3547, 3803, 4079, 4363, 4663, 4987, 5309, 5647, 5953, 6311, 6689, 7027, 7481, 7841, 8263, 8689

Offset: 0

Author

Cino Hilliard, Dec 20 2002

Keywords

• nonn
• easy

Comments

Primes on the left side of the triangle formed by listing successively the prime numbers in a triangular grid:

2

3 5

7 11 13

17 19 23 29

31 37 41 43 47

53 59 61 67 71 73

The sum of the reciprocals appears to converge.

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

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

Examples

			    2     3     7    17    31    53    79   109   157   199
    5    11    19    37    59    83   113   163   211   269
   13    23    41    61    89   127   167   223   271   347
   29    43    67    97   131   173   227   277   349   421
   47    71   101   137   179   229   281   353   431   503
   73   103   139   181   233   283   359   433   509   607
  107   149   191   239   293   367   439   521   613   709
  151   193   241   307   373   443   523   617   719   827
  197   251   311   379   449   541   619   727   829   953
  257   313   383   457   547   631   733   839   967  1087
Interspersion of rows 3 and 5 begins with
  41  61  89  127...
    43  67  97...
Interspersion of columns 3 and 5 begins with
  19
    31
  41
    59
  67
    89
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A011756, A078722, A078723, A078724, A078725, A151800.

Cf. A114537.

Programs

Magma

[NthPrime(n*(n + 1) div 2+1): n in [0..50]]; // Vincenzo Librandi, Jun 08 2016

Mathematica

Table[Prime[n (n + 1)/2 + 1], {n, 0, 46}] (* Michael De Vlieger, Oct 28 2015 *)
Prime[#]&/@(Accumulate[Range[0,50]]+1) (* Harvey P. Dale, Aug 04 2018 *)
Grid[Table[Prime[n + (n + k - 2) (n + k - 1)/2], {n, 1, 20}, {k, 1, 15}]]
(* Clark Kimberling, Apr 13 2025 *)

PARI

a(n) = prime(n*(n+1)/2 + 1);

Formula

a(n) = A000040(A000124(n)). - Altug Alkan, Oct 28 2015

a(n) = A151800(A011756(n)) for n >= 1. - Amiram Eldar, Sep 05 2024