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A238380
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Numbers k such that the average of the divisors of k and k+1 is the same.
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8
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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