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

%I A307908 #21 Oct 12 2024 21:33:54
%S A307908 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,
%T A307908 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,
%U A307908 7,1,7,1,7,4,7,3,7,1,7,4,7,1,7,3,7,5,7
%N A307908 a(n) is the least k such that p^k >= n for any prime factor p of n.
%H A307908 Rémy Sigrist, <a href="/A307908/a307908.png">Colored scatterplot of the ordinal transform of n -> 5*a(n) - a(n^5) for n = 2..100000</a>
%F A307908 a(n) = ceiling(log(n)/log(A020639(n))).
%F A307908 a(p^k) = k for any prime number p and any k > 0.
%F A307908 0 <= k*a(n) - a(n^k) < k for any n > 1 and k > 0.
%F A307908 a(n) = 1 iff n is a prime number (A000040).
%F A307908 a(n) = 2 iff n is the square of a prime number (A001248).
%e A307908 For n = 12:
%e A307908 - the prime factors of 12 are 2 and 3,
%e A307908 - 3^4 > 2^4 >= 12 > 2^3,
%e A307908 - hence a(n) = 4.
%t A307908 Array[If[PrimeQ@ #, 1, Ceiling@ Log[FactorInteger[#][[1, 1]], #]] &, 105, 2] (* _Michael De Vlieger_, May 08 2019 *)
%o A307908 (PARI) a(n) = my (f=factor(n)); logint(n, f[1,1]) + if (#f~>1, 1, 0)
%o A307908 (Python)
%o A307908 from operator import sub
%o A307908 from sympy import integer_log, primefactors
%o A307908 def A307908(n): return 1+sub(*integer_log(n,min(primefactors(n)))) # _Chai Wah Wu_, Oct 12 2024
%Y A307908 Cf. A000040, A001248, A020639, A307907.
%K A307908 nonn
%O A307908 2,3
%A A307908 _Rémy Sigrist_, May 05 2019