A333267 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) * k_j), where pi = A000720.

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 2, 1, 4, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 3, 4, 1, 1, 1, 5, 1, 2, 2, 4, 2, 3, 1, 3, 1, 2, 2, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 3, 1, 6, 3, 1, 2, 2, 2, 1, 4, 6, 1, 1, 3, 4, 2, 2, 2, 6, 2, 2, 2, 6, 2, 1, 1, 4, 4, 1, 2, 4, 2, 2, 1, 3, 3, 2, 2, 4, 1, 1, 3, 5, 2, 4, 2, 4

Offset: 1

Author

Ilya Gutkovskiy, Mar 13 2020

Keywords

• nonn
• mult

Examples

			a(36) = a(2^2 * 3^2) = a(prime(1)^2 * prime(2)^2) = a(1) * 2 * a(2) * 2 = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000026, A000720, A002110, A003963, A005361, A054725, A109129, A276625 (positions of 1's), A282446, A304117, A318046, A328880.

Programs

Maple

a:= proc(n) option remember;
      mul(a(numtheory[pi](i[1]))*i[2], i=ifactors(n)[2])
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 13 2020

Mathematica

a[1] = 1; a[n_] := a[n] = Times @@ (a[PrimePi[#[[1]]]] #[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]

Formula

a(n) = A005361(n) * Product_{p|n, p prime} a(pi(p)).

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

a(p^k) = k * a(p), where p is prime.

a(A002110(n)) = Product_{k=1..n} a(k).