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.

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 16 2020

STATUS

approved