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A238380 Numbers k such that the average of the divisors of k and k+1 is the same. 8
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 (list; graph; refs; listen; history; text; internal format)
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 terms 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] &]

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: A198091 A197797 A224245 * A183307 A334547 A348777

Adjacent sequences:  A238377 A238378 A238379 * A238381 A238382 A238383

KEYWORD

nonn

AUTHOR

Giovanni Resta, Feb 25 2014

STATUS

approved

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Last modified December 3 22:32 EST 2021. Contains 349468 sequences. (Running on oeis4.)