

A331382


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


8



1, 2, 4, 8, 16, 18, 20, 32, 35, 44, 60, 62, 64, 65, 68, 72, 92, 95, 98, 128, 154, 160, 168, 256, 264, 288, 291, 303, 324, 364, 400, 476, 480, 512, 618, 623, 624, 642, 706, 763, 791, 812, 816, 826, 938, 994, 1024, 1036, 1064, 1068, 1106, 1144, 1148, 1152, 1162
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OFFSET

1,2


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

Table of n, a(n) for n=1..55.


EXAMPLE

The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
8: {1,1,1}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
32: {1,1,1,1,1}
35: {3,4}
44: {1,1,5}
60: {1,1,2,3}
62: {1,11}
64: {1,1,1,1,1,1}
65: {3,6}
68: {1,1,7}
72: {1,1,1,2,2}
92: {1,1,9}
95: {3,8}
98: {1,4,4}
128: {1,1,1,1,1,1,1}
For example, 60 has prime factors {2,2,3,5} and prime indices {1,1,2,3}, and 12 is divisible by 6, so 60 is in the sequence.


MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Divisible[Plus@@Prime/@primeMS[#], Times@@primeMS[#]]&]


CROSSREFS

These are the Heinz numbers of the partitions counted by A331381.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to the sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379.
Sequence in context: A018366 A216781 A127066 * A154362 A226221 A072462
Adjacent sequences: A331379 A331380 A331381 * A331383 A331384 A331385


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jan 16 2020


STATUS

approved



