A340607 Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 2, 0, 1, 3, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 2, 0, 1, 4

Offset

1,27

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Example

The a(n) factorizations for n = 27, 84, 108, 180, 252, 360, 432:
  27     2*6*7   2*6*9      4*5*9      4*7*9      5*8*9       6*8*9
  3*3*3  3*4*7   3*4*9      2*2*45     6*6*7      2*4*45      2*8*27
         2*2*21  2*2*27     2*6*15     2*2*63     3*8*15      4*4*27
                 2*2*3*3*3  3*4*15     2*6*21     4*6*15      2*2*2*6*9
                            2*2*3*3*5  3*4*21     2*12*15     2*2*3*4*9
                                       2*2*3*3*7  2*2*2*5*9   2*2*2*2*27
                                                  2*3*3*4*5   2*2*2*2*3*3*3
                                                  2*2*2*3*15

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[n], OddQ[Length[#]]&&OddQ[Max@@#]&]], {n, 100}]

Prog

(PARI) A340607(n, m=n, k=0, grodd=0) = if(1==n, k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(grodd||(d%2)), s += A340607(n/d, d, 1-k, bitor(1, grodd)))); (s)); \\ Antti Karttunen, Dec 13 2021

Crossrefs

Note: Heinz numbers are given in parentheses below.

The case of odd length only is A339890.

The case of all odd factors is A340102.

The version for partitions is A340385.

The version for prime indices is A340386.

The case of odd maximum only is A340831.

A000009 counts partitions into odd parts (A066208).

A001055 counts factorizations, with strict case A045778.

A027193 counts partitions of odd length/maximum (A026424/A244991).

A058695 counts partitions of odd numbers (A300063).

A078408 counts odd-length partitions into odd numbers (A300272).

A316439 counts factorizations by sum and length.

A340101 counts factorizations (into odd factors = of odd numbers).

A340832 counts factorizations whose least part is odd.

A340854/A340855 lack/have a factorization with odd minimum.

Cf. A000700, A024429, A026804, A028260, A061395, A112798, A160786, A236914, A324522, A326845, A340608, A340788.

Sequence in context: A037908 A116663 A258940 * A319659 A050372 A037802

Adjacent sequences: A340604 A340605 A340606 * A340608 A340609 A340610

Keyword

nonn

Author

Gus Wiseman, Jan 25 2021

Extensions

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

Status

approved