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.

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

Links

Table of n, a(n) for n=1..71.

Wikipedia, Geometric mean

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.

Cf. A000041, A102627, A320322, A326028, A326641.

Sequence in context: A057022 A287896 A087504 * A166312 A138099 A359634

Adjacent sequences: A067536 A067537 A067538 * A067540 A067541 A067542

Keyword

easy,nonn

Author

Naohiro Nomoto, Jan 27 2002

Extensions

Terms a(61) onwards from Max Alekseyev, Feb 06 2010

Status

approved