A326622 Number of factorizations of n into factors > 1 with integer average.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 5, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 3, 3, 1, 1, 6, 2, 2, 2, 2, 1, 2, 2, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 1, 1, 3, 2, 1, 1, 6, 1, 1, 3, 2, 2, 2, 1, 7, 5, 1, 1, 4, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 8, 1, 1, 3, 3, 1, 1, 1, 4, 5, 1, 1, 6

Offset

1,4

Links

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

Example

The a(80) = 7 factorizations:
  (2*2*2*10)
  (2*2*20)
  (2*5*8)
  (2*40)
  (4*20)
  (8*10)
  (80)

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], IntegerQ[Mean[#]]&]], {n, 2, 100}]

Prog

(PARI) A326622(n, m=n, facsum=0, facnum=0) = if(1==n, facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

Crossrefs

Partitions with integer average are A067538.

Strict partitions with integer average are A102627.

Heinz numbers of partitions with integer average are A316413.

Factorizations with integer geometric mean are A326028.

Cf. A001055, A051293, A078174, A078175, A326514, A326515, A326567/A326568, A326621, A326623, A326667 (= a(2^n)), A327906 (positions of 1's), A327907 (of terms > 1).

Sequence in context: A305193 A038538 A293515 * A292777 A088529 A267116

Adjacent sequences: A326619 A326620 A326621 * A326623 A326624 A326625

Keyword

nonn

Author

Gus Wiseman, Jul 14 2019

Extensions

Data section extended up to a(108), with missing term a(1)=0 also added (thus correcting the offset) - Antti Karttunen, Nov 10 2024

Status

approved