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

%I A307907 #22 Oct 12 2024 21:33:49
%S A307907 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,
%T A307907 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,
%U A307907 2,1,3,1,1,2,1,1,1,1,2,4,1,1,2,1,1,1,1
%N A307907 a(n) is the greatest k such that p^k <= n for any prime factor p of n.
%H A307907 Rémy Sigrist, <a href="/A307907/a307907.png">Colored scatterplot of the ordinal transform of n -> a(n^5) - 5*a(n) for n = 2..100000</a>
%F A307907 a(n) = floor(log(n)/log(A006530(n))).
%F A307907 a(p^k) = k for any prime number p and any k > 0.
%F A307907 0 <= a(n^k) - k*a(n) < k for any n > 1 and any k > 0.
%F A307907 a(n) = 1 iff n belongs to A064052.
%F A307907 a(n) > 1 iff n belongs to A048098.
%F A307907 a(n) > 2 iff n belongs to A090081.
%e A307907 For n = 12:
%e A307907 - the prime factors of 12 are 2 and 3,
%e A307907 - 2^2 < 3^2 <= 12 < 3^3,
%e A307907 - hence a(12) = 2.
%t A307907 Array[If[PrimeQ@ #, 1, Floor@ Log[FactorInteger[#][[-1, 1]], #]] &, 105, 2] (* _Michael De Vlieger_, May 08 2019 *)
%o A307907 (PARI) a(n) = my (f=factor(n)); logint(n, f[#f~, 1])
%o A307907 (Python)
%o A307907 from sympy import integer_log, primefactors
%o A307907 def A307907(n): return integer_log(n,max(primefactors(n)))[0] # _Chai Wah Wu_, Oct 12 2024
%Y A307907 Cf. A006530, A048098, A064052, A090081, A307908.
%K A307907 nonn
%O A307907 2,3
%A A307907 _Rémy Sigrist_, May 05 2019