cp's OEIS Frontend

A331378 Numbers whose product of prime indices is divisible by their sum of prime factors.

%I A331378 #12 Feb 07 2021 06:25:38
%S A331378 35,65,95,98,154,189,297,324,363,364,375,450,476,585,623,702,763,765,
%T A331378 791,812,826,918,938,994,1036,1064,1106,1144,1148,1162,1197,1225,1287,
%U A331378 1288,1300,1305,1309,1449,1470,1484,1517,1566,1593,1665,1708,1710,1736,1769
%N A331378 Numbers whose product of prime indices is divisible by their sum of prime factors.
%C A331378 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.
%H A331378 Amiram Eldar, <a href="/A331378/b331378.txt">Table of n, a(n) for n = 1..10000</a>
%e A331378 The sequence of terms together with their prime indices begins:
%e A331378     35: {3,4}
%e A331378     65: {3,6}
%e A331378     95: {3,8}
%e A331378     98: {1,4,4}
%e A331378    154: {1,4,5}
%e A331378    189: {2,2,2,4}
%e A331378    297: {2,2,2,5}
%e A331378    324: {1,1,2,2,2,2}
%e A331378    363: {2,5,5}
%e A331378    364: {1,1,4,6}
%e A331378    375: {2,3,3,3}
%e A331378    450: {1,2,2,3,3}
%e A331378    476: {1,1,4,7}
%e A331378    585: {2,2,3,6}
%e A331378    623: {4,24}
%e A331378    702: {1,2,2,2,6}
%e A331378    763: {4,29}
%e A331378    765: {2,2,3,7}
%e A331378    791: {4,30}
%e A331378    812: {1,1,4,10}
%e A331378 For example, 450 = prime(1)*prime(2)*prime(2)*prime(3)*prime(3) has prime indices {1,2,2,3,3} and prime factors {2,3,3,5,5}, and since 36 is divisible by 18, 450 is in the sequence.
%t A331378 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331378 Select[Range[2,1000],Divisible[Times@@primeMS[#],Total[Prime/@primeMS[#]]]&]
%Y A331378 These are the Heinz numbers of the partitions counted by A330954.
%Y A331378 Numbers divisible by the sum of their prime factors are A036844.
%Y A331378 Numbers divisible by the sum of their prime indices are A324851.
%Y A331378 Sum of prime indices divides product of prime indices: A326149.
%Y A331378 Partitions whose Heinz number is divisible by their sum are A330950.
%Y A331378 Partitions whose product divides their sum of primes are A331381.
%Y A331378 Product of prime indices equals sum of prime factors: A331384.
%Y A331378 Cf. A000040, A001414, A056239, A057568, A324850, A330953, A331379, A331381, A331382, A331383.
%K A331378 nonn
%O A331378 1,1
%A A331378 _Gus Wiseman_, Jan 15 2020