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A383158 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

%I A383158 #7 Apr 20 2025 02:39:40
%S A383158 1,2,2,1,2,4,2,2,1,4,2,6,2,4,4,1,2,6,2,6,4,4,2,8,1,4,2,6,2,8,2,2,4,4,
%T A383158 4,9,2,4,4,8,2,8,2,6,6,4,2,10,1,6,4,6,2,8,4,8,4,4,2,4,2,4,6,1,4,8,2,6,
%U A383158 4,8,2,6,2,4,6,6,4,8,2,10,1,4,2,4,4,4,4
%N A383158 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the divisors of n.
%C A383158 a(n) depends only on the prime signature of n (A118914).
%H A383158 Amiram Eldar, <a href="/A383158/b383158.txt">Table of n, a(n) for n = 1..10000</a>
%F A383158 a(n) = denominator(Sum_{d|n} A051903(d) / A000005(n)) = denominator(A383156(n) / A000005(n)).
%F A383158 a(A056798(n)) = 1. a(n) = 1 also for other numbers: 1800, 2700, 3528, ...
%e A383158 Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 7/6, ...
%e A383158 4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = denominator((0 + 1 + 2)/3) = denominator(1) = 1.
%e A383158 12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = denominator((0 + 1 + 1 + 2 + 1 + 2)/6) = denominator(7/6) = 6.
%t A383158 emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[DivisorSum[n, emax[#] &] / DivisorSigma[0, n]]; Array[a, 100]
%o A383158 (PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
%o A383158 a(n) = my(f = factor(n)); denominator(sumdiv(n, d, emax(d)) / numdiv(f));
%Y A383158 Cf. A000005, A051903, A056798, A118914, A383156, A383157 (numerators).
%K A383158 nonn,easy,frac
%O A383158 1,2
%A A383158 _Amiram Eldar_, Apr 18 2025