A340786 Number of factorizations of 4n into an even number of even factors > 1.

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

Maple

g:= proc(n, m, p)
 option remember;
 local F, r, x, i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)), list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n], EvenQ[Length[#]]&&Select[#, OddQ]=={}&]], {n, 100}]

Prog

(PARI)
A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

Crossrefs

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

Positions of ones are 1 and A000040, or A008578.

A version for partitions is A027187 (A028260).

Allowing odd length gives A108501 (bisection of A340785).

Allowing odd factors gives A339846.

An odd version is A340102.

- Factorizations -

A001055 counts factorizations, with strict case A045778.

A316439 counts factorizations by product and length.

A340101 counts factorizations into odd factors.

A340653 counts balanced factorizations.

A340831/A340832 count factorizations with odd maximum/minimum.

- Even -

A027187 counts partitions of even maximum (A244990).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A236913 counts partitions of even length and sum (A340784).

A340601 counts partitions of even rank (A340602).

Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

Sequence in context: A252477 A029351 A178638 * A290496 A035115 A370897

Adjacent sequences: A340783 A340784 A340785 * A340787 A340788 A340789

Keyword

nonn

Author

Gus Wiseman, Jan 31 2021

Status

approved