A240914 Semiprimes of the form S(n) + T(n) where S(n) and T(n) are the n-th square and the n-th triangular numbers.

15, 26, 57, 77, 155, 187, 301, 551, 737, 1027, 1457, 1751, 3197, 3337, 5251, 6767, 7597, 8251, 13301, 22387, 24257, 25807, 32047, 34277, 41417, 41917, 48151, 61307, 63757, 66887, 68801, 85801, 103097, 112751, 136957, 141527, 145237, 179747, 180787, 196747

Offset: 1

Author

K. D. Bajpai, Apr 14 2014

Keywords

• nonn

Comments

The n-th triangular number T(n) = n/2*(n+1).

Semiprimes (biprimes) in the sequence are product of two primes, simultaneously sum of n-th square & triangular numbers.

All the terms in the sequence, except a(2), are odd numbers.

Examples

			a(1) = 15: 3^2 + 3/2*(3+1) = 15 = 3*5, which is product of two primes. Hence it is semiprime.
a(3) = 57: 6^2 + 6/2*(6+1) = 57 = 3*19, which is product of two primes. Hence it is semiprime.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..9010

Crossrefs

Cf. A001358, A005898, A046388, A240859, A240884.

Programs

Maple

with(numtheory):KD:= proc() local a,b; a:=(n)^2 + n/2*(n+1);b:=bigomega(a); if b=2 then RETURN (a); fi; end: seq(KD(), n=1..500);

Mathematica

KD = {}; Do[t = n^2 + n/2*(n + 1); If[PrimeOmega[t] == 2, AppendTo[KD, t]], {n, 500}]; KD
c=0; Do[t=n^2 + n/2*(n+1); If[PrimeOmega[t]==2,c=c+1; Print[c," ",t]], {n,1,500000}];
Module[{nn=500,s,t},s=Range[nn]^2;t=Accumulate[Range[nn]];Select[ Total/@ Thread[{s,t}],PrimeOmega[#]==2&]] (* or *) Select[ Table[ (n(1+3n))/2,{n,500}],PrimeOmega[#]==2&](* Harvey P. Dale, Feb 07 2018 *)