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A326624 Heinz numbers of non-constant integer partitions whose geometric mean is an integer.

%I A326624 #5 Jul 15 2019 01:45:04
%S A326624 14,42,46,57,76,106,126,161,183,185,194,196,228,230,302,371,378,393,
%T A326624 399,412,424,454,477,515,588,622,679,684,687,722,742,781,786,838,1057,
%U A326624 1064,1077,1082,1115,1134,1150,1157,1159,1219,1244,1272,1322,1563,1589,1654
%N A326624 Heinz numbers of non-constant integer partitions whose geometric mean is an integer.
%C A326624 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A326624 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326624 The sequence of terms together with their prime indices begins:
%e A326624     14: {1,4}
%e A326624     42: {1,2,4}
%e A326624     46: {1,9}
%e A326624     57: {2,8}
%e A326624     76: {1,1,8}
%e A326624    106: {1,16}
%e A326624    126: {1,2,2,4}
%e A326624    161: {4,9}
%e A326624    183: {2,18}
%e A326624    185: {3,12}
%e A326624    194: {1,25}
%e A326624    196: {1,1,4,4}
%e A326624    228: {1,1,2,8}
%e A326624    230: {1,3,9}
%e A326624    302: {1,36}
%e A326624    371: {4,16}
%e A326624    378: {1,2,2,2,4}
%e A326624    393: {2,32}
%e A326624    399: {2,4,8}
%e A326624    412: {1,1,27}
%t A326624 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326624 Select[Range[100],!PrimePowerQ[#]&&IntegerQ[GeometricMean[primeMS[#]]]&]
%Y A326624 The case with prime powers is A326623.
%Y A326624 Subsets whose geometric mean is an integer are A326027.
%Y A326624 Cf. A001055, A067539, A078175, A102627, A316413, A326567/A326568, A326622, A326625.
%K A326624 nonn
%O A326624 1,1
%A A326624 _Gus Wiseman_, Jul 14 2019