A375080 a(n) is the numerator of ( Sum_{d|n} (n - d) )/tau(n).

0, 1, 1, 5, 2, 3, 3, 17, 14, 11, 5, 22, 6, 8, 9, 49, 8, 23, 9, 13, 13, 13, 11, 33, 44, 31, 17, 56, 14, 21, 15, 43, 21, 41, 23, 233, 18, 23, 25, 115, 20, 30, 21, 30, 32, 28, 23, 178, 30, 69, 33, 107, 26, 39, 37, 41, 37, 71, 29, 46, 30, 38, 137, 321, 44, 48, 33, 47, 45, 52

Offset: 1

Author

Stefano Spezia, Jul 29 2024

Keywords

• nonn
• frac

Comments

( Sum_{d|n} (n - d) )/tau(n) is the average distance between n and its divisor.

Crossrefs

Cf. A000005, A000203, A057020, A057021 (denominator).

Programs

Mathematica

a[n_]:=Numerator[n-DivisorSigma[1,n]/DivisorSigma[0,n]];  Array[a,70]

Python

from math import prod
from fractions import Fraction
from sympy import factorint
def A375080(n):
    f = factorint(n).items()
    return (n-Fraction(prod((p**(e+1)-1)//(p-1) for p, e in f),prod(e+1 for p,e in f))).numerator # Chai Wah Wu, Jul 30 2024

Formula

a(n) = numerator((n - sigma(n))/tau(n)).

a(n) = numerator(n - A000203(n)/A000005(n)).

a(n) = numerator(n - A057020(n)/A057021(n)).