A108502 -id:A108502

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A108501

Number of factorizations of 4*n into even numbers.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

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

FORMULA

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

EXAMPLE

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

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jun 06 2005

STATUS

approved

A108503

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

+10
4

2, 4, 3, 8, 3, 8, 3, 16, 4, 8, 3, 20, 3, 8, 5, 32, 3, 13, 3, 20, 5, 8, 3, 48, 4, 8, 5, 20, 3, 18, 3, 64, 5, 8, 5, 38, 3, 8, 5, 48, 3, 18, 3, 20, 7, 8, 3, 112, 4, 13, 5, 20, 3, 19, 5, 48, 5, 8, 3, 56, 3, 8, 7, 128, 5, 18, 3, 20, 5, 18, 3, 104, 3, 8, 7, 20, 5, 18, 3, 112, 6, 8, 3, 56, 5, 8, 5, 48

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

a(9)=4 because 9*4=36 can be factored as 36=18*2=2*18=6*6.

MAPLE

with(numtheory):

b:= proc(n) option remember; 1+add(`if`(irem(d, 2)=0 and

irem(n/d, 2)=0, b(d), 0), d=divisors(n) minus {1, n})

end:

a:= n-> b(4*n):

seq(a(n), n=1..100); # Alois P. Heinz, Feb 17 2015

MATHEMATICA

nn = 300; f[list_, i] := list[[i]]; a = Table[Boole[EvenQ[n]], {n, 1, nn}]; Select[Map[Total, Transpose[NestList[Table[DirichletConvolve[f[#, n], f[a, n], n, m], {m, 1, nn}] &, a, nn]]], # > 1 &] (* Geoffrey Critzer, Feb 17 2015 *)

CROSSREFS

Cf. A074206, A108501, A108502.

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jun 06 2005

STATUS

approved

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