A240884 Semiprimes of the form C(n) + T(n) where C(n) and T(n) are the n-th cube and triangular numbers.

33, 74, 237, 371, 1055, 1397, 10901, 12443, 30287, 39899, 55613, 80453, 207149, 303041, 360467, 407999, 639797, 1230821, 1650053, 2056511, 2695349, 2873441, 3454427, 3956873, 9823349, 10384103, 13680599, 15844877, 16419449, 20608499, 22705373, 26508143

Offset: 1

Author

K. D. Bajpai, Apr 14 2014

Keywords

• nonn
• easy

Comments

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

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

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

Examples

			a(1) = 33: 3^3 + 3/2*(3+1) = 33 = 3*11, which is product of two primes and hence semiprime.
a(3) = 237: 6^3 + 6/2*(6+1) = 237 = 3*79, which is product of two primes and hence semiprime.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..6556

Crossrefs

Cf. A001358, A005898, A046388.

Programs

Maple

with(numtheory):KD:= proc() local a,b; a:=(n)^3+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^3 + n/2*(n + 1); If[PrimeOmega[t] == 2, AppendTo[KD, t]], {n, 500}]; KD

PARI

has(n)=if(n%2, isprime(n) && isprime(n^2+n\2+1), isprime(n/2) && isprime(2*n^2+n+1))
for(n=1,1e4, if(has(n), print1(n^3+n*(n+1)/2", "))) \\ Charles R Greathouse IV, Aug 25 2014