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A326149 Numbers whose product of prime indices is divisible by their sum of prime indices. 17

%I

%S 2,3,5,7,9,11,13,17,19,23,29,30,31,37,41,43,47,49,53,59,61,63,65,67,

%T 71,73,79,81,83,84,89,97,101,103,107,108,109,113,125,127,131,137,139,

%U 149,150,151,154,157,163,165,167,169,173,179,181,190,191,193,197

%N Numbers whose product of prime indices is divisible by their sum 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.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is divisible by their sum of parts. The enumeration of these partitions by sum is given by A057568.

%e The sequence of terms together with their prime indices begins:

%e 2: {1}

%e 3: {2}

%e 5: {3}

%e 7: {4}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 17: {7}

%e 19: {8}

%e 23: {9}

%e 29: {10}

%e 30: {1,2,3}

%e 31: {11}

%e 37: {12}

%e 41: {13}

%e 43: {14}

%e 47: {15}

%e 49: {4,4}

%e 53: {16}

%e 59: {17}

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

%t Select[Range[2,100],Divisible[Times@@primeMS[#],Plus@@primeMS[#]]&]

%Y Satisfies A056239(a(n))|A003963(a(n)).

%Y The nonprime case is A326150, with squarefree case A326158.

%Y Cf. A000720, A001222, A057567, A057568, A112798, A301987.

%Y Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 09 2019

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Last modified June 4 03:40 EDT 2020. Contains 334815 sequences. (Running on oeis4.)