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A229469 Numbers k such that T(k) + S(k) + 1 is prime, where T(k) and S(k) are the k-th triangular and square numbers.

%I A229469 #16 Jul 28 2025 20:17:40
%S A229469 1,5,8,9,12,17,21,24,29,32,41,44,45,53,56,57,60,68,69,77,81,84,89,92,
%T A229469 96,108,113,117,120,132,144,149,156,164,185,197,200,201,212,213,224,
%U A229469 233,236,248,252,260,264,269,281,288,300,312,317,321,324,329,344,353
%N A229469 Numbers k such that T(k) + S(k) + 1 is prime, where T(k) and S(k) are the k-th triangular and square numbers.
%H A229469 K. D. Bajpai, <a href="/A229469/b229469.txt">Table of n, a(n) for n = 1..10000</a>
%e A229469 a(4)=9: T(9)+S(9)+1= 9/2*(9+1)+9^2+1= 127 which is prime.
%e A229469 a(5)=12: T(12)+S(12)+1= 12/2*(12+1)+12^2+1= 223 which is prime.
%p A229469 KD:= proc() local a,b,c,d; a:= n/2*(n+1)+n^2+1; if isprime(a) then RETURN(n): fi; end: seq(KD(),n=1..5000);
%o A229469 (PARI) v=List(); for(n=1, 10^5, if(isprime(n/2*(n+1)+n^2+1), listput(v, n))); Vec(v)
%o A229469 (PARI) is(n)=isprime(n*(3*n+1)/2+1) \\ _Charles R Greathouse IV_, Sep 24 2013
%Y A229469 Cf. A000217, A000290.
%Y A229469 Cf. A228908, A229157.
%K A229469 nonn,easy
%O A229469 1,2
%A A229469 _K. D. Bajpai_, Sep 24 2013