This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A067539 #12 Sep 24 2023 13:04:35
%S A067539 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,
%T A067539 26,19,24,20,30,28,32,27,37,35,48,37,45,37,51,51,58,50,64,62,83,73,84,
%U A067539 69,91,89,101,97,116,111,136,123,142,138,160,161,181,171,205,199,231,221
%N 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.
%C A067539 a(n) is the number of integer partitions of n whose geometric mean is an integer. - _Gus Wiseman_, Jul 19 2019
%H A067539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A067539 From _Gus Wiseman_, Jul 19 2019: (Start)
%e A067539 The a(1) = 1 through a(8) = 4 partitions:
%e A067539 (1) (2) (3) (4) (5) (6) (7) (8)
%e A067539 (11) (111) (22) (41) (33) (421) (44)
%e A067539 (1111) (11111) (222) (1111111) (2222)
%e A067539 (111111) (11111111)
%e A067539 (End)
%t A067539 Table[Length[Select[IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,30}] (* _Gus Wiseman_, Jul 19 2019 *)
%o A067539 (Python)
%o A067539 from math import prod
%o A067539 from sympy import integer_nthroot
%o A067539 from sympy.utilities.iterables import partitions
%o A067539 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
%Y A067539 Partitions with integer average are A067538.
%Y A067539 Subsets whose geometric mean is an integer are A326027.
%Y A067539 The Heinz numbers of these partitions are A326623.
%Y A067539 The strict case is A326625.
%Y A067539 Cf. A000041, A102627, A320322, A326028, A326641.
%K A067539 easy,nonn
%O A067539 1,2
%A A067539 _Naohiro Nomoto_, Jan 27 2002
%E A067539 Terms a(61) onwards from _Max Alekseyev_, Feb 06 2010