%I #5 May 17 2019 22:06:56
%S 1,9,25,49,77,121,125,169,221,245,289,323,343,361,375,437,529,841,899,
%T 961,1331,1369,1517,1681,1763,1849,1859,2021,2197,2209,2401,2773,2809,
%U 2873,3127,3481,3721,3757,4087,4489,4757,4913,5041,5183,5329,5929,6137,6241
%N Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of 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 9: {2,2}
%e 25: {3,3}
%e 49: {4,4}
%e 77: {4,5}
%e 121: {5,5}
%e 125: {3,3,3}
%e 169: {6,6}
%e 221: {6,7}
%e 245: {3,4,4}
%e 289: {7,7}
%e 323: {7,8}
%e 343: {4,4,4}
%e 361: {8,8}
%e 375: {2,3,3,3}
%e 437: {8,9}
%e 529: {9,9}
%e 841: {10,10}
%e 899: {10,11}
%e 961: {11,11}
%e For example, the sequence contains 245 because the prime indices of 245 are {3,4,4}, with reciprocal sum 1/6 + 1/24 + 1/24 = 1/4.
%t Select[Range[1000],!PrimeQ[#]&&IntegerQ[1/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, A316854, A316857, A325618, A325620, A325622, A325623.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 17 2019