A308993 Multiplicative with a(p) = 1 and a(p^e) = p^a(e) for any e > 1 and prime number p.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Rémy Sigrist, Jul 04 2019

Keywords

• nonn
• mult

Comments

To compute a(n): remove every prime number at leaf position in the prime tower factorization of n (the prime tower factorization of a number is defined in A182318).

Examples

			See Links sections.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Illustration of first terms

Crossrefs

Cf. A004709, A005117, A182318, A185102.

Programs

PARI

a(n) = my (f=factor(n)); prod (i=1, #f~, f[i,1]^if (f[i,2]==1, 0, a(f[i,2])))

Formula

a(n) = 1 iff n is squarefree.

a^k(n) = 1 for any k >= A185102(n) (where a^k denotes the k-th iterate of a).

a(n)^2 <= n with equality iff n is the square of some cubefree number (n = A004709(k)^2 for some k > 0).