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A326645 Heinz numbers of integer partitions whose mean and geometric mean are both integers.

%I A326645 #9 Jul 16 2019 22:02:16
%S A326645 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,46,47,49,53,
%T A326645 57,59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,
%U A326645 128,131,137,139,149,151,157,163,167,169,173,179,181,183,191,193
%N A326645 Heinz numbers of integer partitions whose mean and geometric mean are both integers.
%C A326645 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326645 The enumeration of these partitions by sum is given by A326641.
%H A326645 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326645 The sequence of terms together with their prime indices begins:
%e A326645     2: {1}
%e A326645     3: {2}
%e A326645     4: {1,1}
%e A326645     5: {3}
%e A326645     7: {4}
%e A326645     8: {1,1,1}
%e A326645     9: {2,2}
%e A326645    11: {5}
%e A326645    13: {6}
%e A326645    16: {1,1,1,1}
%e A326645    17: {7}
%e A326645    19: {8}
%e A326645    23: {9}
%e A326645    25: {3,3}
%e A326645    27: {2,2,2}
%e A326645    29: {10}
%e A326645    31: {11}
%e A326645    32: {1,1,1,1,1}
%e A326645    37: {12}
%e A326645    41: {13}
%e A326645    43: {14}
%e A326645    46: {1,9}
%e A326645    47: {15}
%e A326645    49: {4,4}
%t A326645 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326645 Select[Range[100],IntegerQ[Mean[primeMS[#]]]&&IntegerQ[GeometricMean[primeMS[#]]]&]
%Y A326645 Heinz numbers of partitions with integer mean are A316413.
%Y A326645 Heinz numbers of partitions with integer geometric mean are A326623.
%Y A326645 Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
%Y A326645 Partitions with integer mean and geometric mean are A326641.
%Y A326645 Subsets with integer mean and geometric mean are A326643.
%Y A326645 Strict partitions with integer mean and geometric mean are A326029.
%Y A326645 Cf. A051293, A056239, A067538, A067539, A078175, A112798, A316413, A326623, A326644, A326646, A326647.
%K A326645 nonn
%O A326645 1,1
%A A326645 _Gus Wiseman_, Jul 16 2019