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.
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}
324: {1,1,2,2,2,2}
364: {1,1,4,6}
476: {1,1,4,7}
623: {4,24}
763: {4,29}
791: {4,30}
812: {1,1,4,10}
826: {1,4,17}
938: {1,4,19}
994: {1,4,20}
1036: {1,1,4,12}
1064: {1,1,1,4,8}
1106: {1,4,22}
1144: {1,1,1,5,6}
1148: {1,1,4,13}
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.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Times@@primeMS[#]==Plus@@Prime/@primeMS[#]&]
CROSSREFS
These are the Heinz numbers of the partitions counted by A331383.
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.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2020
STATUS
approved