A307907 a(n) is the greatest k such that p^k <= n for any prime factor p of n.

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

Offset: 2

Author

Rémy Sigrist, May 05 2019

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A006530, A048098, A064052, A090081, A307908.

Programs

Mathematica

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

PARI

a(n) = my (f=factor(n)); logint(n, f[#f~, 1])

Python

from sympy import integer_log, primefactors
def A307907(n): return integer_log(n,max(primefactors(n)))[0] # Chai Wah Wu, Oct 12 2024

Formula

a(n) = floor(log(n)/log(A006530(n))).

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

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

a(n) = 1 iff n belongs to A064052.

a(n) > 1 iff n belongs to A048098.

a(n) > 2 iff n belongs to A090081.