A333235 a(n) is the product of indices of unitary prime power divisors of n.

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset: 1

Author

Ilya Gutkovskiy, Mar 12 2020

Keywords

• nonn
• mult
• look

Comments

Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - M. F. Hasler, Jun 16 2021

Examples

			a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

Programs

Maple

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

Mathematica

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

PARI

apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

Formula

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).

a(prime(n)) = A027883(n).

a(2^n) = A182908(n).

a(A246655(n)) = n.