A326027 - OEIS

A326027

Number of nonempty subsets of {1..n} whose geometric mean is an integer.

0, 1, 2, 3, 6, 7, 8, 9, 12, 19, 20, 21, 28, 29, 30, 31, 40, 41, 70, 71, 74, 75, 76, 77, 108, 123, 124, 211, 214, 215, 216, 217, 332, 333, 334, 335, 592, 593, 594, 595, 612, 613, 614, 615, 618, 639, 640, 641, 1160, 1183, 1324, 1325, 1328, 1329, 2176, 2177, 2196, 2197, 2198, 2199, 2414, 2415, 2416, 2443, 4000, 4001, 4002, 4003, 4006, 4007, 4008, 4009, 6626, 6627, 6628, 9753, 9756, 9757, 9758, 9759, 11136

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OFFSET

0,3

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..255

Wikipedia, Geometric mean

FORMULA

a(n) = A357413(n) + A357414(n). For a squarefree n, a(n) = a(n-1) + 1. - Max Alekseyev, Mar 01 2025

EXAMPLE

The a(1) = 1 through a(9) = 19 subsets:

{1} {1} {1} {1} {1} {1} {1} {1} {1}

{2} {2} {2} {2} {2} {2} {2} {2}

{3} {3} {3} {3} {3} {3} {3}

{4} {4} {4} {4} {4} {4}

{1,4} {5} {5} {5} {5} {5}

{1,2,4} {1,4} {6} {6} {6} {6}

{1,2,4} {1,4} {7} {7} {7}

{1,2,4} {1,4} {8} {8}

{1,2,4} {1,4} {9}

{2,8} {1,4}

{1,2,4} {1,9}

{2,4,8} {2,8}

{4,9}

{1,2,4}

{1,3,9}

{2,4,8}

{3,8,9}

{4,6,9}

{3,6,8,9}

MATHEMATICA

Table[Length[Select[Subsets[Range[n]], IntegerQ[GeometricMean[#]]&]], {n, 0, 10}]

CROSSREFS

First differences are A082553.

Partitions whose geometric mean is an integer are A067539.

Strict partitions whose geometric mean is an integer are A326625.

Subsets whose average is an integer are A051293.

Cf. A078174, A078175, A102627, A326567/A326568, A326622, A326623, A326624.

Sequence in context: A003605 A344128 A132188 * A255527 A316156 A319737

Adjacent sequences: A326024 A326025 A326026 * A326028 A326029 A326030

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 14 2019

EXTENSIONS

Terms a(57) onward from Max Alekseyev, Mar 01 2025

STATUS

approved