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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A139566 A097963 A063936 * A205881 A132452 A063552

Adjacent sequences: A240911 A240912 A240913 * A240915 A240916 A240917

Keyword

nonn

Author

K. D. Bajpai, Apr 14 2014

Status

approved