OFFSET
1,1
COMMENTS
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:
2: {1}
3: {2}
5: {3}
7: {4}
9: {2,2}
11: {5}
13: {6}
17: {7}
19: {8}
23: {9}
29: {10}
31: {11}
36: {1,1,2,2}
MATHEMATICA
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n, red[n]]&&suQ[red[n]];
Select[Range[2, 200], suQ[#]&]
CROSSREFS
The first term that is not a prime power A000961 is 36.
The first term that is not a perfect power A001597 is 1260.
Removing all primes gives A353389.
These partitions are counted by A353426.
The version for compositions is A353431.
A003963 gives product of prime indices.
A325131 lists numbers relatively prime to their prime shadow.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
STATUS
approved