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

0,20

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

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A001222, A001358, A072931.

Sequence in context: A068914 A090824 A264620 * A277264 A259538 A099314

Adjacent sequences: A302298 A302299 A302300 * A302302 A302303 A302304

Keyword

nonn,look,easy

Author

Wesley Ivan Hurt, Apr 04 2018

Status

approved