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

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, 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, 4, 8, 2, 1, 2, 4, 4, 4, 4, 8, 2, 4, 1, 4, 2, 8, 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, gcd(d, n/d) = 1} A051903(d) / A034444(n)) = denominator(A383159(n) / A034444(n)).

a(A056798(n)) = 1. a(n) = 1 also for other numbers: 72, 108, 200, 288, 392, 500, 675, 800, 968, 972, ...

Example

Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 5/4, ...
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.
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.

Mathematica

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Denominator[DivisorSum[n, emax[#] &, CoprimeQ[#, n/#] &] / 2^PrimeNu[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) * (gcd(d, n/d) == 1)) / (1 << omega(f)));

Crossrefs

The unitary version of A383158.

Cf. A034444, A051903, A056798, A077610, A118914, A383159, A383160 (numerators).

Sequence in context: A366308 A347089 A354825 * A383158 A210531 A066954

Adjacent sequences: A383158 A383159 A383160 * A383162 A383163 A383164

Keyword

nonn,easy,frac

Author

Amiram Eldar, Apr 18 2025

Status

approved