A326623 - OEIS

%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