A159047 Primes which are triangular numbers plus 3.

3, 13, 31, 139, 193, 409, 499, 823, 1381, 1543, 2083, 2281, 3163, 3919, 6673, 7753, 9319, 9733, 17581, 19309, 21739, 22369, 24979, 27031, 27733, 30631, 39343, 40189, 51043, 53959, 54949, 57973, 62131, 67531, 70879, 81409, 85081, 86323, 91381

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

Keywords

• nonn

Comments

For n>1, a(n)== 1 (mod 6). [Proof: the triangular numbers are {0,1,3,4} (mod 6), see A104686. 3 plus triangular numbers in the same set, and only those == 1 (mod 6) can be primes.] - Zak Seidov, Oct 16 2015

Examples

			13=10+3, 31=28+3, 139=136+3, 193=190+3, 409=406+3, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217, A055469, A055472.

Programs

Mathematica

s=0;lst={};Do[s+=n;p=s+3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Table[n*(n + 1)/2 + 3, {n, 0, 250}], PrimeQ] (* G. C. Greubel, Jul 13 2017 *)
Select[Accumulate[Range[0,500]]+3,PrimeQ] (* Harvey P. Dale, Jul 30 2018 *)

PARI

for(n=0, 1e3, if(isprime(k=3+n*(n+1)/2), print1(k", "))) \\ Altug Alkan, Oct 16 2015