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%I #15 Jul 30 2018 17:36:03
%S 3,13,31,139,193,409,499,823,1381,1543,2083,2281,3163,3919,6673,7753,
%T 9319,9733,17581,19309,21739,22369,24979,27031,27733,30631,39343,
%U 40189,51043,53959,54949,57973,62131,67531,70879,81409,85081,86323,91381
%N Primes which are triangular numbers plus 3.
%C 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
%H G. C. Greubel, <a href="/A159047/b159047.txt">Table of n, a(n) for n = 1..1000</a>
%e 13=10+3, 31=28+3, 139=136+3, 193=190+3, 409=406+3, ...
%t s=0;lst={};Do[s+=n;p=s+3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
%t Select[Table[n*(n + 1)/2 + 3, {n, 0, 250}], PrimeQ] (* _G. C. Greubel_, Jul 13 2017 *)
%t Select[Accumulate[Range[0,500]]+3,PrimeQ] (* _Harvey P. Dale_, Jul 30 2018 *)
%o (PARI) for(n=0, 1e3, if(isprime(k=3+n*(n+1)/2), print1(k", "))) \\ _Altug Alkan_, Oct 16 2015
%Y Cf. A000217, A055469, A055472.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Apr 03 2009
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