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Heinz numbers of integer partitions whose mean and geometric mean are both integers.
14

%I #9 Jul 16 2019 22:02:16

%S 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 57,59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,

%U 128,131,137,139,149,151,157,163,167,169,173,179,181,183,191,193

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

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C The enumeration of these partitions by sum is given by A326641.

%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 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}

%e 41: {13}

%e 43: {14}

%e 46: {1,9}

%e 47: {15}

%e 49: {4,4}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],IntegerQ[Mean[primeMS[#]]]&&IntegerQ[GeometricMean[primeMS[#]]]&]

%Y Heinz numbers of partitions with integer mean are A316413.

%Y Heinz numbers of partitions with integer geometric mean are A326623.

%Y Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.

%Y Partitions with integer mean and geometric mean are A326641.

%Y Subsets with integer mean and geometric mean are A326643.

%Y Strict partitions with integer mean and geometric mean are A326029.

%Y Cf. A051293, A056239, A067538, A067539, A078175, A112798, A316413, A326623, A326644, A326646, A326647.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 16 2019