A364691 Pentagonal numbers which are the sum of the first k primes, for some k >= 0.

0, 5, 13490, 3299391550, 22042432252064127, 2387505511919644051, 680588297594638712735

Offset: 1

Author

Paolo Xausa, Aug 03 2023

Keywords

• nonn
• hard
• more

Examples

			5 is a term because it's both a pentagonal number and the sum of the first two primes (2 + 3).
		

Links

• Wikipedia, Pentagonal number.
• Index to sequences related to polygonal numbers

Crossrefs

Intersection of A000326 with A007504.

Cf. A066527, A061890, A364694, A364696, A366269.

Programs

Mathematica

A364691list[kmax_]:=Module[{p=0},Join[{0},Table[If[IntegerQ[(Sqrt[24(p+=Prime[k])+1]+1)/6],p,Nothing],{k,kmax}]]];A364691list[25000] (* Paolo Xausa, Oct 06 2023 *)

PARI

ispenta(n) = my(s); issquare(24*n+1,&s)&&s%6==5;
my(S=0); forprime (p=2, oo, S+=p; if (ispenta(S), print1(S,", "))) \\ Hugo Pfoertner, Aug 03 2023

Extensions

a(5)-a(7) from Hugo Pfoertner, Aug 04 2023