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

1,1

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.

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002961.

Sequence in context: A197797 A224245 A374510 * A183307 A334547 A348777

Adjacent sequences: A238377 A238378 A238379 * A238381 A238382 A238383

Keyword

nonn

Author

Giovanni Resta, Feb 25 2014

Status

approved