A108501 Number of factorizations of 4*n into even numbers.

2, 3, 2, 5, 2, 4, 2, 7, 3, 4, 2, 7, 2, 4, 3, 11, 2, 6, 2, 7, 3, 4, 2, 12, 3, 4, 3, 7, 2, 7, 2, 15, 3, 4, 3, 12, 2, 4, 3, 12, 2, 7, 2, 7, 4, 4, 2, 19, 3, 6, 3, 7, 2, 8, 3, 12, 3, 4, 2, 14, 2, 4, 4, 22, 3, 7, 2, 7, 3, 7, 2, 21, 2, 4, 4, 7, 3, 7, 2, 19, 4, 4, 2, 14, 3, 4, 3, 12, 2, 11, 3, 7, 3, 4, 3, 30, 2

Offset: 1

Author

Christian G. Bower, Jun 06 2005

Keywords

• nonn

Comments

a(n) = 2 iff n is 1 or an odd prime (A006005); in this case, the two factorizations are 4n = 2 * 2n. - Bernard Schott, Nov 30 2020

Examples

			a(6)=4 because 6*4=24 can be factored as 24=12*2=6*4=6*2*2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A001055, A006005, A108502, A108503.

Programs

Maple

with(numtheory):
b:= proc(n, i) option remember; `if`(n<=i, 1, 0)+
      add(`if`(d<=i and irem(d, 2)=0 and irem(n/d, 2)=0,
      b(n/d, min(d, i)), 0), d=divisors(n) minus {1, n})
    end:
a:= n-> b(4*n$2):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 17 2015

Mathematica

b[n_, i_] := b[n, i] = If[n <= i, 1, 0] + Sum[If[d <= i && Mod[d, 2]==0 && Mod[n/d, 2]==0, b[n/d, Min[d, i]], 0], {d, Divisors[n][[2 ;; -2]]}];
a[n_] := b[4n, 4n];
Array[a, 100] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

Formula

a(2^k) = A000041(k+2). - Bernard Schott, Dec 01 2020