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A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

%I A326623 #7 Jul 15 2019 01:44:56
%S A326623 2,3,4,5,7,8,9,11,13,14,16,17,19,23,25,27,29,31,32,37,41,42,43,46,47,
%T A326623 49,53,57,59,61,64,67,71,73,76,79,81,83,89,97,101,103,106,107,109,113,
%U A326623 121,125,126,127,128,131,137,139,149,151,157,161,163,167,169
%N A326623 Heinz numbers of integer partitions whose geometric mean is an integer.
%C A326623 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A326623 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326623 The sequence of terms together with their prime indices begins:
%e A326623     2: {1}
%e A326623     3: {2}
%e A326623     4: {1,1}
%e A326623     5: {3}
%e A326623     7: {4}
%e A326623     8: {1,1,1}
%e A326623     9: {2,2}
%e A326623    11: {5}
%e A326623    13: {6}
%e A326623    14: {1,4}
%e A326623    16: {1,1,1,1}
%e A326623    17: {7}
%e A326623    19: {8}
%e A326623    23: {9}
%e A326623    25: {3,3}
%e A326623    27: {2,2,2}
%e A326623    29: {10}
%e A326623    31: {11}
%e A326623    32: {1,1,1,1,1}
%e A326623    37: {12}
%t A326623 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326623 Select[Range[100],IntegerQ[GeometricMean[primeMS[#]]]&]
%Y A326623 The enumeration of these partitions by sum is given by A067539.
%Y A326623 Heinz numbers of partitions with integer average are A316413.
%Y A326623 The case without prime powers is A326624.
%Y A326623 Subsets whose geometric mean is an integer are A326027.
%Y A326623 Factorizations with integer geometric mean are A326028.
%Y A326623 Cf. A001055, A078175, A102627, A326567/A326568, A326622, A326625.
%K A326623 nonn
%O A326623 1,1
%A A326623 _Gus Wiseman_, Jul 14 2019