A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 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, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

First differs from A294336 and A316782 at a(36) = 5.

Links

Table of n, a(n) for n=1..88.

Wikipedia, Geometric mean

Formula

a(2^n) = A067538(n).

Example

The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
  (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)
  (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)
                         (4*4)                                   (6*6)
                         (2*2*2*2)                               (2*18)
                                                                 (3*12)

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[GeometricMean[#]]&]], {n, 2, 100}]

Crossrefs

Positions of terms > 1 are the perfect powers A001597.

Partitions with integer geometric mean are A067539.

Subsets with integer geometric mean are A326027.

Factorizations with integer average and geometric mean are A326647.

Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

Sequence in context: A294336 A316782 A326647 * A294338 A316790 A316789

Adjacent sequences: A326025 A326026 A326027 * A326029 A326030 A326031

Keyword

nonn

Author

Gus Wiseman, Jul 15 2019

Status

approved