A359910 Number of odd-length integer factorizations of n into factors > 1 with the same mean as median.

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

Offset

1,8

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

The a(n) factorizations for n = 120, 960, 5760, 6720:
  120      960         5760            6720
  4*5*6    2*16*30     16*18*20        4*30*56
  2*6*10   4*12*20     3*5*6*8*8       10*21*32
           8*10*12     4*4*6*6*10      12*20*28
           3*4*4*4*5   2*2*8*10*18     4*5*6*7*8
                       2*2*2*4*4*5*9   2*4*7*10*12
                                       2*2*2*4*5*6*7

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[#]]&&Mean[#]==Median[#]&]], {n, 100}]

Prog

(PARI) A359910(n, m=n, facs=List([])) = if(1==n, (((#facs)%2) && (facs[(1+#facs)/2]==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A359910(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Crossrefs

The version for partitions is A359895, ranked by A359891.

This is the odd-length case of A359909, partitions A240219.

A001055 counts factorizations.

A326622 counts factorizations with integer mean, strict A328966.

Cf. A316313, A326567/A326568, A359889, A359894, A359897, A359902, A359906, A359911, A360005.

Sequence in context: A307427 A380395 A318672 * A368168 A359411 A367516

Adjacent sequences: A359907 A359908 A359909 * A359911 A359912 A359913

Keyword

nonn

Author

Gus Wiseman, Jan 24 2023

Extensions

More terms from Antti Karttunen, Jan 20 2025

Status

approved