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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 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: A198091 A197797 A224245 * A183307 A334547 A348777
KEYWORD
nonn
AUTHOR
Giovanni Resta, Feb 25 2014
STATUS
approved

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Last modified March 4 22:06 EST 2024. Contains 370532 sequences. (Running on oeis4.)