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%I #6 May 13 2019 08:12:17
%S 1,2,4,8,9,16,18,32,36,64,72,81,128,144,162,256,288,324,375,512,576,
%T 648,729,750,1024,1152,1296,1458,1500,2048,2304,2592,2916,3000,3375,
%U 4096,4608,5184,5832,6000,6561,6750,8192,9216,10368,11664,12000,13122,13500
%N Heinz numbers of integer partitions whose reciprocal factorial sum is an integer.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.
%e The sequence of terms together with their prime indices begins:
%e 1: {}
%e 2: {1}
%e 4: {1,1}
%e 8: {1,1,1}
%e 9: {2,2}
%e 16: {1,1,1,1}
%e 18: {1,2,2}
%e 32: {1,1,1,1,1}
%e 36: {1,1,2,2}
%e 64: {1,1,1,1,1,1}
%e 72: {1,1,1,2,2}
%e 81: {2,2,2,2}
%e 128: {1,1,1,1,1,1,1}
%e 144: {1,1,1,1,2,2}
%e 162: {1,2,2,2,2}
%e 256: {1,1,1,1,1,1,1,1}
%e 288: {1,1,1,1,1,2,2}
%e 324: {1,1,2,2,2,2}
%e 375: {2,3,3,3}
%e 512: {1,1,1,1,1,1,1,1,1}
%t Select[Range[1000],IntegerQ[Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]
%Y Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
%Y Reciprocal factorial sum: A002966, A058360, A316856, A325619, A325620, A325623.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 13 2019
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