cp's OEIS Frontend

A383161 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.

%I A383161 #8 Apr 20 2025 02:38:27
%S A383161 1,2,2,1,2,4,2,2,1,4,2,4,2,4,4,1,2,4,2,4,4,4,2,4,1,4,2,4,2,8,2,2,4,4,
%T A383161 4,2,2,4,4,4,2,8,2,4,4,4,2,4,1,4,4,4,2,4,4,4,4,4,2,8,2,4,4,1,4,8,2,4,
%U A383161 4,8,2,1,2,4,4,4,4,8,2,4,1,4,2,8,4,4,4,4
%N A383161 a(n) is the denominator of the mean of the maximum exponents in the prime factorizations of the unitary divisors of n.
%C A383161 a(n) depends only on the prime signature of n (A118914).
%H A383161 Amiram Eldar, <a href="/A383161/b383161.txt">Table of n, a(n) for n = 1..10000</a>
%F A383161 a(n) = denominator(Sum_{d|n, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = denominator(A383159(n) / A034444(n)).
%F A383161 a(A056798(n)) = 1. a(n) = 1 also for other numbers: 72, 108, 200, 288, 392, 500, 675, 800, 968, 972, ...
%e A383161 Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 5/4, ...
%e A383161 4 has 2 unitary divisors: 1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0 and 2, respectively. Therefore, a(4) = denominator((0 + 2)/2) = denominator(1) = 1.
%e A383161 12 has 4 unitary divisors: 1, 3 = 3^1, 4 = 2^2 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 2 and 2, respectively. Therefore, a(12) = denominator((0 + 1 + 2 + 2)/4) = denominator(5/4) = 4.
%t A383161 emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &] / 2^PrimeNu[n]]; Array[a, 100]
%o A383161 (PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
%o A383161 a(n) = my(f = factor(n)); denominator(sumdiv(n, d, emax(d) * (gcd(d, n/d) == 1)) / (1 << omega(f)));
%Y A383161 The unitary version of A383158.
%Y A383161 Cf. A034444, A051903, A056798, A077610, A118914, A383159, A383160 (numerators).
%K A383161 nonn,easy,frac
%O A383161 1,2
%A A383161 _Amiram Eldar_, Apr 18 2025