A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

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, 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, 1, 7, 3, 7, 1, 11, 1, 3, 7, 7, 3, 7, 1, 21, 2, 3, 1, 5, 3

Offset: 1

Author

Amiram Eldar, Apr 18 2025

Keywords

• nonn
• easy
• frac

Comments

a(n) depends only on the prime signature of n (A118914).

Examples

			Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 7/6, ...
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.
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383158(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).

Crossrefs

Cf. A000005, A001248, A051903, A118914, A308043, A345231, A361062, A383156, A383158 (denominators).

Programs

Mathematica

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &] / DivisorSigma[0, n]]; Array[a, 100]

PARI

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d)) / numdiv(f));

Formula

a(n) = numerator(Sum_{d|n} A051903(d) / A000005(n)) = numerator(A383156(n) / A000005(n)).

a(n)/A383158(n) = 1 if and only if n is a square of a prime (A001248).

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... .