login
A331378
Numbers whose product of prime indices is divisible by their sum of prime factors.
15
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
1,1
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.
LINKS
EXAMPLE
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.
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[#]]]&]
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.
Sequence in context: A176875 A292005 A338008 * A331384 A297150 A292081
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
STATUS
approved