A108503 -id:A108503 - cp's OEIS frontend

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

Christian G. Bower, Jun 06 2005

nonn

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

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

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

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

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

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 *)

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

A108502 Number of factorizations of 4*n into distinct even numbers.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 3, 4, 2, 4, 2, 5, 3, 3, 2, 7, 2, 3, 3, 5, 2, 6, 2, 5, 3, 3, 3, 7, 2, 3, 3, 7, 2, 6, 2, 5, 4, 3, 2, 10, 2, 4, 3, 5, 2, 6, 3, 7, 3, 3, 2, 11, 2, 3, 4, 6, 3, 6, 2, 5, 3, 6, 2, 11, 2, 3, 4, 5, 3, 6, 2, 10, 3, 3, 2, 11, 3, 3, 3, 7, 2, 9, 3, 5, 3, 3, 3, 14, 2, 4, 4, 7, 2, 6

Offset: 1

Christian G. Bower, Jun 06 2005

nonn

			a(15)=3 because 15*4=60 can be factored as 60=30*2=10*6.

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

Cf. A045778, A108501, A108503.

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-1, 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

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-1, 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 *)

A352063 Number of ordered factorizations of 2*n + 1 into odd factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 4, 1, 1, 3, 3, 1, 3, 1, 1, 8, 1, 2, 3, 1, 3, 3, 1, 1, 8, 3, 1, 3, 1, 1, 8, 3, 1, 8, 1, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 13, 1, 1, 3, 1, 3, 8, 3, 2, 3, 4, 1, 3, 1, 3, 20, 1, 1, 3, 3, 3, 8, 1, 1, 8, 3, 1, 3, 3, 1, 13, 1, 2, 8, 1, 8, 3, 1, 1, 3, 3, 3, 20, 1, 1, 13, 1, 1, 3, 3, 3, 8, 3, 1

Offset: 0

Ilya Gutkovskiy, Mar 04 2022

nonn

Also number of perfect partitions of 2*n.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Perfect Partition
Eric Weisstein's World of Mathematics, Ordered Factorization

Cf. A002033, A058696, A064216, A074206, A108503, A340101, A349907, A378222.

A074206[1] = 1; A074206[n_] := A074206[n] = Sum[If[d < n, A074206[d], 0], {d, Divisors[n]}]; a[n_] := A074206[2 n + 1]; Table[a[n], {n, 0, 90}]

A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A352063(n) = A074206(1+n+n); \\ Antti Karttunen, Nov 24 2024

a(n) = A002033(2*n) = A074206(2*n+1).

a(n) = A378222(2*n+1) = A074206(A064216(1+n)). - Antti Karttunen, Nov 24 2024

More terms from Antti Karttunen, Nov 24 2024

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

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A108502 Number of factorizations of 4*n into distinct even numbers.

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A352063 Number of ordered factorizations of 2*n + 1 into odd factors > 1.

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