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Number of factorizations of 2n into even factors > 1.
15

%I #16 Dec 14 2021 00:24:30

%S 1,2,1,3,1,2,1,5,1,2,1,4,1,2,1,7,1,3,1,4,1,2,1,7,1,2,1,4,1,3,1,11,1,2,

%T 1,6,1,2,1,7,1,3,1,4,1,2,1,12,1,3,1,4,1,3,1,7,1,2,1,7,1,2,1,15,1,3,1,

%U 4,1,3,1,12,1,2,1,4,1,3,1,12,1,2,1,7,1

%N Number of factorizations of 2n into even factors > 1.

%H Antti Karttunen, <a href="/A340785/b340785.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A349906(2*n). - _Antti Karttunen_, Dec 13 2021

%e The a(n) factorizations for n = 2*2, 2*4, 2*8, 2*12, 2*16, 2*32, 2*36, 2*48 are:

%e 4 8 16 24 32 64 72 96

%e 2*2 2*4 2*8 4*6 4*8 8*8 2*36 2*48

%e 2*2*2 4*4 2*12 2*16 2*32 4*18 4*24

%e 2*2*4 2*2*6 2*2*8 4*16 6*12 6*16

%e 2*2*2*2 2*4*4 2*4*8 2*6*6 8*12

%e 2*2*2*4 4*4*4 2*2*18 2*6*8

%e 2*2*2*2*2 2*2*16 4*4*6

%e 2*2*2*8 2*2*24

%e 2*2*4*4 2*4*12

%e 2*2*2*2*4 2*2*4*6

%e 2*2*2*2*2*2 2*2*2*12

%e 2*2*2*2*6

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Select[#,OddQ]=={}&]],{n,2,100,2}]

%o (PARI)

%o A349906(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A349906(n/d, d))); (s));

%o A340785(n) = A349906(2*n); \\ _Antti Karttunen_, Dec 13 2021

%Y Note: A-numbers of Heinz-number sequences are in parentheses below.

%Y The version for partitions is A035363 (A066207).

%Y The odd version is A340101.

%Y The even length case is A340786.

%Y - Factorizations -

%Y A001055 counts factorizations, with strict case A045778.

%Y A340653 counts balanced factorizations.

%Y A340831/A340832 count factorizations with odd maximum/minimum.

%Y A316439 counts factorizations by product and length

%Y A340102 counts odd-length factorizations of odd numbers into odd factors.

%Y - Even -

%Y A027187 counts partitions of even length/maximum (A028260/A244990).

%Y A058696 counts partitions of even numbers (A300061).

%Y A067661 counts strict partitions of even length (A030229).

%Y A236913 counts partitions of even length and sum.

%Y A340601 counts partitions of even rank (A340602).

%Y Cf. A001147, A001222, A050320, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

%Y Even bisection of A349906.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 30 2021