OFFSET
1,3
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.
FORMULA
EXAMPLE
We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.
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]];
Table[Times@@red/@primeMS[n], {n, 100}]
CROSSREFS
Positions of first appearances are A353397.
A003963 gives product of prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
STATUS
approved