A199331 Number of ordered ways of writing n as the sum of two semiprimes.

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 2, 2, 2, 2, 0, 3, 4, 3, 2, 0, 2, 4, 4, 2, 2, 3, 4, 5, 6, 4, 0, 2, 6, 6, 4, 2, 6, 6, 4, 5, 8, 7, 4, 2, 8, 10, 6, 5, 2, 5, 6, 4, 10, 6, 4, 4, 10, 12, 12, 2, 6, 10, 6, 7, 8, 9, 6, 5, 12, 14, 10, 6, 6, 7, 8, 7, 10, 10, 6, 4, 14, 14

Offset: 1

Author

Robert G. Wilson v, Nov 05 2011

Keywords

• easy
• nonn

Comments

Conjecture: Only the integers 1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22, 33 (A072966) cannot be partitioned into a set of two semiprimes.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A035026, A064911, A072966, A073610.

Programs

Maple

sp:= select(t -> numtheory:-bigomega(t)=2, [$1..100]):
G:= expand(add(x^t,t=sp)^2):
seq(coeff(G,x,i),i=1..100); # Robert Israel, Nov 24 2020

Mathematica

mx=200; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; t = Select[ Range@ mx, semiPrimeQ]; s = Sort[Plus @@@ Tuples[t, 2]]; Transpose[ Tally@ Join[ Range@ mx, s]][[2]] - 1

Formula

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A064911. - Wesley Ivan Hurt, Jan 07 2024

Extensions

Definition clarified by Robert Israel, Nov 24 2020