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.
We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
12: {1,1,2}
20: {1,1,3}
36: {1,1,2,2}
44: {1,1,5}
56: {1,1,1,4}
68: {1,1,7}
100: {1,1,3,3}
124: {1,1,11}
164: {1,1,13}
184: {1,1,1,9}
208: {1,1,1,1,6}
236: {1,1,17}
240: {1,1,1,1,2,3}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100], Times@@red/@primeMS[#]==Times@@Last/@FactorInteger[#]&]
CROSSREFS
The partitions are counted by A353398.
Taking indices instead of exponents on the LHS gives A353503.
A325131 lists numbers relatively prime to their prime shadow.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393
- recursive version counted by A353426
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
STATUS
approved