A325130 - OEIS (original) (raw)

1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the integer partitions counted by A276429.

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - Amiram Eldar, Jan 09 2021

EXAMPLE

The sequence of terms together with their prime indices begins:

1: {}

3: {2}

4: {1,1}

5: {3}

7: {4}

8: {1,1,1}

11: {5}

12: {1,1,2}

13: {6}

15: {2,3}

16: {1,1,1,1}

17: {7}

19: {8}

20: {1,1,3}

21: {2,4}

23: {9}

24: {1,1,1,2}

25: {3,3}

27: {2,2,2}

28: {1,1,4}

MAPLE

q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):

a:= proc(n) option remember; local k; for k from 1+

`if`(n=1, 0, a(n-1)) while not q(k) do od; k

end:

MATHEMATICA

Select[Range[100], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k!=PrimePi[p]]&]