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A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

%I A375932 #7 Sep 03 2024 01:17:16
%S A375932 1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,2,1,5,1,1,1,3,1,1,1,7,1,1,1,1,1,1,
%T A375932 1,1,1,1,1,5,1,1,1,11,5,1,1,3,1,2,1,13,1,2,1,7,1,1,1,15,1,1,7,1,1,1,1,
%U A375932 17,1,1,1,9,1,1,3,19,1,1,1,5,1,1,1,21,1
%N A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.
%C A375932 The product of the prime powers in the prime factorization of n that have an exponent that is smaller than the maximum exponent in this factorization.
%H A375932 Amiram Eldar, <a href="/A375932/b375932.txt">Table of n, a(n) for n = 1..10000</a>
%F A375932 If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i < max_{j} e_j]), where [] is the Iverson bracket.
%F A375932 a(n) = n / A375931(n).
%F A375932 a(n) = 1 if and only if n is a power of a squarefree number (A072774).
%F A375932 A051903(a(n)) = A375933(n).
%F A375932 a(n!) = A049606(n) for n != 3.
%e A375932 60 = 2^2 * 3 * 5, and the maximum exponent in the prime factorization of 60 is 2, which is the exponent of its prime factor 2. Therefore a(60) = 3 * 5 = 15.
%t A375932 a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; n / (Times @@ p[[i]])^m]; Array[a, 100]
%o A375932 (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] < m, p[i]^e[i], 1)));}
%Y A375932 Cf. A005117, A049606, A051903, A072774, A375931, A375933.
%K A375932 nonn,easy
%O A375932 1,12
%A A375932 _Amiram Eldar_, Sep 03 2024