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proposed

allocated for Gus WisemanNumber of strict factorizations of n with integer average.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 1, 5, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 5, 1, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 5, 1, 1, 3, 2, 2, 2, 1, 5, 2, 1, 1, 4, 2, 1, 2, 3

2,7

The a(n) factorizations for n = 2, 8, 24, 48, 96:

(2) (8) (24) (32) (48) (96)

(2*4) (4*6) (4*8) (6*8) (2*48)

(2*12) (2*16) (2*24) (4*24)

(2*3*4) (4*12) (6*16)

(2*4*6) (8*12)

(3*4*8)

(2*3*16)

(2*4*12)

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

The non-strict version is A326622.

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, A326515, A326567/A326568, A326619/A326620, A326621, A326625.

allocated

nonn

Gus Wiseman, Nov 16 2019

approved

editing

allocated for Gus Wiseman

allocated

approved