A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 4, 2, 2, 2, 6, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 7, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 4, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 4, 1, 4, 5, 2, 1, 6, 1, 4, 2, 5, 1, 4, 2, 3, 3, 2, 2, 11

Offset

1,4

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 = 24, 36, 60, 120, 144, 360:
  24      36        60      120       144       360
  3*8     4*9       2*30    2*60      2*72      4*90
  4*6     6*6       3*20    3*40      3*48      5*72
  2*12    2*18      4*15    4*30      4*36      6*60
  2*3*4   3*12      5*12    5*24      6*24      8*45
          2*2*3*3   6*10    6*20      8*18      9*40
                    3*4*5   8*15      9*16      10*36
                            10*12     12*12     12*30
                            4*5*6     2*2*6*6   15*24
                            2*6*10    3*3*4*4   18*20
                            2*3*4*5             2*180
                                                3*120
                                                2*10*18
                                                3*4*5*6

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

Prog

(PARI)
median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359909(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A359909(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Crossrefs

The version for partitions is A240219, complement A359894.

These multisets are ranked by A359889.

The version for strict partitions is A359897.

The odd-length case is A359910.

The complement is counted by A359911.

A001055 counts factorizations.

A058398 counts partitions by mean, see also A008284, A327482.

A326622 counts factorizations with integer mean, strict A328966.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A316313, A326567/A326568, A359906, A360005.

Sequence in context: A001222 A257091 A351418 * A319269 A320888 A296132

Adjacent sequences: A359906 A359907 A359908 * A359910 A359911 A359912

Keyword

nonn

Author

Gus Wiseman, Jan 24 2023

Extensions

More terms from Antti Karttunen, Jan 20 2025

Status

approved