This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383157 #10 Apr 20 2025 02:39:14
%S A383157 0,1,1,1,1,3,1,3,1,3,1,7,1,3,3,2,1,7,1,7,3,3,1,13,1,3,3,7,1,7,1,5,3,3,
%T A383157 3,13,1,3,3,13,1,7,1,7,7,3,1,21,1,7,3,7,1,13,3,13,3,3,1,5,1,3,7,3,3,7,
%U A383157 1,7,3,7,1,11,1,3,7,7,3,7,1,21,2,3,1,5,3
%N A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.
%C A383157 a(n) depends only on the prime signature of n (A118914).
%H A383157 Amiram Eldar, <a href="/A383157/b383157.txt">Table of n, a(n) for n = 1..10000</a>
%H A383157 Amiram Eldar, <a href="/A383157/a383157.jpg">Plot of (Sum_{k=1..n} a(k)/A383158(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10)</a>.
%F A383157 a(n) = numerator(Sum_{d|n} A051903(d) / A000005(n)) = numerator(A383156(n) / A000005(n)).
%F A383157 a(n)/A383158(n) = 1 if and only if n is a square of a prime (A001248).
%F A383157 Sum_{k=1..n} a(k)/A383158(k) ~ c_1 * n - c_2 * n /sqrt(log(n)), where c_1 = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.27968644485944694957... is the asymptotic mean of the fractions a(k)/A383158(k), m(k) = Product_{p prime} (1 + (1-1/p) * Sum_{i>=k} (k/(i+1) - 1)/p^i is the asymptotic mean of the ratio between the number of k-free divisors and the number of divisors, e.g., m(2) = A308043 and m(3) = A361062, and c_2 = A345231 = 0.54685595528047446684... .
%e A383157 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 A383157 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) = numerator((0 + 1 + 2)/3) = numerator(1) = 1.
%e A383157 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) = numerator((0 + 1 + 1 + 2 + 1 + 2)/6) = numerator(7/6) = 7.
%t A383157 emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &] / DivisorSigma[0, n]]; Array[a, 100]
%o A383157 (PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
%o A383157 a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d)) / numdiv(f));
%Y A383157 Cf. A000005, A001248, A051903, A118914, A308043, A345231, A361062, A383156, A383158 (denominators).
%K A383157 nonn,easy,frac
%O A383157 1,6
%A A383157 _Amiram Eldar_, Apr 18 2025