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

%I A108501 #21 Dec 02 2020 03:17:55
%S A108501 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,
%T A108501 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,
%U A108501 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
%N A108501 Number of factorizations of 4*n into even numbers.
%C A108501 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
%H A108501 Alois P. Heinz, <a href="/A108501/b108501.txt">Table of n, a(n) for n = 1..10000</a>
%F A108501 a(2^k) = A000041(k+2). - _Bernard Schott_, Dec 01 2020
%e A108501 a(6)=4 because 6*4=24 can be factored as 24=12*2=6*4=6*2*2.
%p A108501 with(numtheory):
%p A108501 b:= proc(n, i) option remember; `if`(n<=i, 1, 0)+
%p A108501       add(`if`(d<=i and irem(d, 2)=0 and irem(n/d, 2)=0,
%p A108501       b(n/d, min(d, i)), 0), d=divisors(n) minus {1, n})
%p A108501     end:
%p A108501 a:= n-> b(4*n$2):
%p A108501 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 17 2015
%t A108501 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]]}];
%t A108501 a[n_] := b[4n, 4n];
%t A108501 Array[a, 100] (* _Jean-François Alcover_, Nov 05 2020, after _Alois P. Heinz_ *)
%Y A108501 Cf. A000041, A001055, A006005, A108502, A108503.
%K A108501 nonn
%O A108501 1,1
%A A108501 _Christian G. Bower_, Jun 06 2005