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