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

35, 65, 95, 98, 154, 189, 297, 324, 363, 364, 375, 450, 476, 585, 623, 702, 763, 765, 791, 812, 826, 918, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1197, 1225, 1287, 1288, 1300, 1305, 1309, 1449, 1470, 1484, 1517, 1566, 1593, 1665, 1708, 1710, 1736, 1769

Offset: 1

Author

Gus Wiseman, Jan 15 2020

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
    35: {3,4}
    65: {3,6}
    95: {3,8}
    98: {1,4,4}
   154: {1,4,5}
   189: {2,2,2,4}
   297: {2,2,2,5}
   324: {1,1,2,2,2,2}
   363: {2,5,5}
   364: {1,1,4,6}
   375: {2,3,3,3}
   450: {1,2,2,3,3}
   476: {1,1,4,7}
   585: {2,2,3,6}
   623: {4,24}
   702: {1,2,2,2,6}
   763: {4,29}
   765: {2,2,3,7}
   791: {4,30}
   812: {1,1,4,10}
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

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

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

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

Partitions whose product divides their sum of primes are A331381.

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@primeMS[#],Total[Prime/@primeMS[#]]]&]