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

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, 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, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 10, 1, 4, 2, 4, 4, 4, 4

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) = denominator((0 + 1 + 2)/3) = denominator(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) = denominator((0 + 1 + 1 + 2 + 1 + 2)/6) = denominator(7/6) = 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A051903, A056798, A118914, A383156, A383157 (numerators).

Programs

Mathematica

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[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)); denominator(sumdiv(n, d, emax(d)) / numdiv(f));

Formula

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

a(A056798(n)) = 1. a(n) = 1 also for other numbers: 1800, 2700, 3528, ...