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A067539
Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.
32
1, 2, 2, 3, 3, 4, 3, 4, 4, 8, 3, 8, 5, 7, 8, 8, 7, 9, 8, 17, 11, 11, 8, 16, 17, 17, 14, 18, 17, 26, 19, 24, 20, 30, 28, 32, 27, 37, 35, 48, 37, 45, 37, 51, 51, 58, 50, 64, 62, 83, 73, 84, 69, 91, 89, 101, 97, 116, 111, 136, 123, 142, 138, 160, 161, 181, 171, 205, 199, 231, 221
OFFSET
1,2
COMMENTS
a(n) is the number of integer partitions of n whose geometric mean is an integer. - Gus Wiseman, Jul 19 2019
EXAMPLE
From Gus Wiseman, Jul 19 2019: (Start)
The a(1) = 1 through a(8) = 4 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (41) (33) (421) (44)
(1111) (11111) (222) (1111111) (2222)
(111111) (11111111)
(End)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], IntegerQ[GeometricMean[#]]&]], {n, 30}] (* Gus Wiseman, Jul 19 2019 *)
PROG
(Python)
from math import prod
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A067539(n): return sum(1 for s, p in partitions(n, size=True) if integer_nthroot(prod(a**b for a, b in p.items()), s)[1]) # Chai Wah Wu, Sep 24 2023
CROSSREFS
Partitions with integer average are A067538.
Subsets whose geometric mean is an integer are A326027.
The Heinz numbers of these partitions are A326623.
The strict case is A326625.
Sequence in context: A057022 A287896 A087504 * A166312 A138099 A359634
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Jan 27 2002
EXTENSIONS
Terms a(61) onwards from Max Alekseyev, Feb 06 2010
STATUS
approved