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%I #6 Jan 17 2020 10:38:12
%S 35,65,95,98,154,324,364,476,623,763,791,812,826,938,994,1036,1064,
%T 1106,1144,1148,1162,1288,1484,1708,1736,2044,2408,2632,4320,5408,
%U 6688,6974,8000,10208,12623,12701,12779,14144,19624,23144,25784,26048,44176,47696
%N Numbers whose sum of prime factors is equal to their product of prime indices.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%e The sequence of terms together with their prime indices begins:
%e 35: {3,4}
%e 65: {3,6}
%e 95: {3,8}
%e 98: {1,4,4}
%e 154: {1,4,5}
%e 324: {1,1,2,2,2,2}
%e 364: {1,1,4,6}
%e 476: {1,1,4,7}
%e 623: {4,24}
%e 763: {4,29}
%e 791: {4,30}
%e 812: {1,1,4,10}
%e 826: {1,4,17}
%e 938: {1,4,19}
%e 994: {1,4,20}
%e 1036: {1,1,4,12}
%e 1064: {1,1,1,4,8}
%e 1106: {1,4,22}
%e 1144: {1,1,1,5,6}
%e 1148: {1,1,4,13}
%e For example, 476 has prime factors {2,2,7,17) and prime indices {1,1,4,7}, and 2+2+7+17 = 28 = 1*1*4*7, so 476 is in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000],Times@@primeMS[#]==Plus@@Prime/@primeMS[#]&]
%Y These are the Heinz numbers of the partitions counted by A331383.
%Y Numbers divisible by the sum of their prime factors are A036844.
%Y Partitions whose product is divisible by their sum are A057568.
%Y Numbers divisible by the sum of their prime indices are A324851.
%Y Product of prime indices is divisible by sum of prime indices: A326149.
%Y Partitions whose Heinz number is divisible by their sum are A330950.
%Y Partitions whose Heinz number is divisible by their sum of primes: A330953.
%Y Sum of prime factors is divisible by sum of prime indices: A331380
%Y Partitions whose product divides their sum of primes are A331381.
%Y Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 16 2020
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