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

1,2

Comments

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

Links

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

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, ...

Example

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.

Mathematica

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

Prog

(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));

Crossrefs

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

Sequence in context: A347089 A354825 A383161 * A210531 A066954 A144925

Adjacent sequences: A383155 A383156 A383157 * A383159 A383160 A383161

Keyword

nonn,easy,frac

Author

Amiram Eldar, Apr 18 2025

Status

approved