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

1,4

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).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Illustration of first terms

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).

Example

See Links sections.

Prog

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

Crossrefs

Cf. A004709, A005117, A182318, A185102.

Sequence in context: A249739 A249740 A071773 * A000188 A162401 A324924

Adjacent sequences: A308990 A308991 A308992 * A308994 A308995 A308996

Keyword

nonn,mult

Author

Rémy Sigrist, Jul 04 2019

Status

approved