A302301 Number of ways to write n as a sum of two distinct semiprimes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 0, 1, 3, 3, 2, 1, 3, 3, 2, 2, 4, 3, 2, 1, 4, 5, 3, 2, 1, 2, 3, 2, 5, 3, 2, 2, 5, 6, 6, 1, 3, 5, 3, 3, 4, 4, 3, 2, 6, 7, 5, 3, 3, 3, 4, 3, 5, 5, 3, 2, 7, 7, 2, 4

Offset: 0

Author

Wesley Ivan Hurt, Apr 04 2018

Keywords

• nonn
• look
• easy

Examples

			a(19) = 2; 19 = 15+4 = 10+9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1000 terms from Harvey P. Dale)
• Index entries for sequences related to partitions

Crossrefs

Cf. A001222, A001358, A072931.

Programs

Maple

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, x*b(n-i, h(min(n-i, i-1))))+b(n, h(i-1)))), x, 3)
    end:
a:= n-> coeff(b(n, h(n)), x, 2):
seq(a(n), n=0..120);  # Alois P. Heinz, May 26 2021

Mathematica

Table[Sum[KroneckerDelta[PrimeOmega[i], 2] KroneckerDelta[PrimeOmega[n - i], 2], {i, Floor[(n - 1)/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}],?(PrimeOmega[#[[1]]]==PrimeOmega[#[[2]]]==2&&#[[1]]!=#[[2]]&)],{n,90}] (* _Harvey P. Dale, Aug 03 2020 *)

PARI

a(n) = sum(i=1, (n-1)\2, (bigomega(i)==2)*(bigomega(n-i)==2)); \\ Michel Marcus, Apr 08 2018

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} [Omega(i) = 2] * [Omega(n-i) = 2], where Omega = A001222 and [] is the Iverson bracket.