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

Author

Gus Wiseman, Jul 14 2019

Keywords

• nonn

Examples

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

Links

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

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).

Programs

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}]

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

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