A238380 Numbers k such that the average of the divisors of k and k+1 is the same.

5, 14, 91, 1334, 1634, 2685, 3478, 5452, 9063, 13915, 16225, 20118, 20712, 33998, 42818, 47795, 64665, 79338, 84134, 103410, 106144, 109214, 111683, 122073, 123497, 133767, 166934, 170884, 203898, 224561, 228377, 267630, 289454, 383594, 384857, 391348, 440013

Offset: 1

Author

Giovanni Resta, Feb 25 2014

Keywords

• nonn

Comments

The average of the divisors of n is equal to sigma(n)/tau(n).

Up to 5*10^12, there are only 3 terms for which the mean is not an integer, namely 254641594575, 280895287491 and 328966666100.

Examples

			91 is a term since the average of the divisors of 91 and 92 is the same. Indeed, (1+7+13+91)/4 = (1+2+4+23+46+92)/6.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..6934 (terms < 5*10^12)

Crossrefs

Cf. A002961.

Programs

Mathematica

av[n_] := DivisorSigma[1,n]/DivisorSigma[0,n]; Select[Range[10^5], av[#] == av[# + 1] &]
SequencePosition[Table[DivisorSigma[1,n]/DivisorSigma[0,n],{n,450000}],{x_,x_}][[All,1]] (* Harvey P. Dale, Jun 01 2022 *)

Python

from sympy import divisors
from fractions import Fraction
def aupto(limit):
  alst, prev_divavg = [], 1
  for n in range(2, limit+2):
    divs = divisors(n)
    divavg = Fraction(sum(divs), len(divs))
    if divavg == prev_divavg: alst.append(n-1)
    prev_divavg = divavg
  return alst
print(aupto(440013)) # Michael S. Branicky, May 14 2021