A331384 - OEIS

%I #10 Dec 19 2024 21:17:41

%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.

%C Numbers k such that A001414(k) = A003963(k). - _Jason Yuen_, Dec 19 2024

%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, A003963, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

%K nonn,changed

%O 1,1

%A _Gus Wiseman_, Jan 16 2020