A331378 - OEIS

%I #12 Feb 07 2021 06:25:38

%S 35,65,95,98,154,189,297,324,363,364,375,450,476,585,623,702,763,765,

%T 791,812,826,918,938,994,1036,1064,1106,1144,1148,1162,1197,1225,1287,

%U 1288,1300,1305,1309,1449,1470,1484,1517,1566,1593,1665,1708,1710,1736,1769

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

%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.

%H Amiram Eldar, <a href="/A331378/b331378.txt">Table of n, a(n) for n = 1..10000</a>

%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    189: {2,2,2,4}

%e    297: {2,2,2,5}

%e    324: {1,1,2,2,2,2}

%e    363: {2,5,5}

%e    364: {1,1,4,6}

%e    375: {2,3,3,3}

%e    450: {1,2,2,3,3}

%e    476: {1,1,4,7}

%e    585: {2,2,3,6}

%e    623: {4,24}

%e    702: {1,2,2,2,6}

%e    763: {4,29}

%e    765: {2,2,3,7}

%e    791: {4,30}

%e    812: {1,1,4,10}

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

%t Select[Range[2,1000],Divisible[Times@@primeMS[#],Total[Prime/@primeMS[#]]]&]

%Y These are the Heinz numbers of the partitions counted by A330954.

%Y Numbers divisible by the sum of their prime factors are A036844.

%Y Numbers divisible by the sum of their prime indices are A324851.

%Y Sum of prime indices divides product of prime indices: A326149.

%Y Partitions whose Heinz number is divisible by their sum are A330950.

%Y Partitions whose product divides their sum of primes are A331381.

%Y Product of prime indices equals sum of prime factors: A331384.

%Y Cf. A000040, A001414, A056239, A057568, A324850, A330953, A331379, A331381, A331382, A331383.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 15 2020