A307908 a(n) is the least k such that p^k >= n for any prime factor p of n.

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

Offset

2,3

Links

Table of n, a(n) for n=2..88.

Rémy Sigrist, Colored scatterplot of the ordinal transform of n -> 5*a(n) - a(n^5) for n = 2..100000

Formula

a(n) = ceiling(log(n)/log(A020639(n))).

a(p^k) = k for any prime number p and any k > 0.

0 <= k*a(n) - a(n^k) < k for any n > 1 and k > 0.

a(n) = 1 iff n is a prime number (A000040).

a(n) = 2 iff n is the square of a prime number (A001248).

Example

For n = 12:
- the prime factors of 12 are 2 and 3,
- 3^4 > 2^4 >= 12 > 2^3,
- hence a(n) = 4.

Mathematica

Array[If[PrimeQ@ #, 1, Ceiling@ Log[FactorInteger[#][[1, 1]], #]] &, 105, 2] (* Michael De Vlieger, May 08 2019 *)

Prog

(PARI) a(n) = my (f=factor(n)); logint(n, f[1, 1]) + if (#f~>1, 1, 0)

Crossrefs

Cf. A000040, A001248, A020639, A307907.

Sequence in context: A144079 A326839 A071575 * A366737 A316436 A303674

Adjacent sequences: A307905 A307906 A307907 * A307909 A307910 A307911

Keyword

nonn

Author

Rémy Sigrist, May 05 2019

Status

approved