

A342100


Abundant numbers k at which the ratio (number of abundant numbers in 1..k)/k reaches a new record high.


0



12, 18, 20, 24, 40, 42, 56, 60, 72, 80, 84, 88, 90, 102, 104, 108, 112, 114, 354, 366, 368, 372, 380, 384, 392, 396, 400, 402, 464, 468, 476, 480, 492, 500, 504, 552, 560, 564, 572, 576, 580, 582, 650, 654, 836, 840, 945, 948, 952, 954, 1002, 2002, 2004, 2024
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OFFSET

1,1


COMMENTS

Let rho(k) = (number of abundant numbers in 1..k)/k. According to A302991 ("Decimal expansion of the asymptotic density of abundant numbers"), lim_{k>infinity} rho(k) = 0.247619...
a(115) = 7254; rho(7254) = 1810/7254 = 0.2495175075820...
Conjecture: a(115) is the final term of this sequence.


LINKS

Table of n, a(n) for n=1..54.


EXAMPLE

k=12 is the 1st abundant number, so at k=12, rho(k) increases from 0 to 1/12 = 0.08333..., a record high, so a(1)=12.
k=18 is the 2nd abundant number, so at k=18, rho(k) reaches 2/18 = 1/9 = 0.11111..., the next record high, so a(2)=18.
k=20 is the 3rd abundant number, so at k=20, rho(k) reaches 3/20 = 0.15, the next record high, so a(3)=20.
k=24 is the 4th abundant number, so at k=24, rho(k) reaches 4/24 = 1/6 = 0.16666..., the next record high, so a(4)=24.
k=30 is the 5th abundant number, so at k=30, rho(k) again reaches 5/30 = 1/6; this is not a new record high, so 30 is not a term of the sequence.


MATHEMATICA

s = {}; c = 0; rm = 0; Do[If[DivisorSigma[1, n] > 2*n, c++; If[(r = c/n) > rm, rm = r; AppendTo[s, n]]], {n, 1, 10^4}]; s (* Amiram Eldar, Feb 28 2021 *)


CROSSREFS

Cf. A005101 (abundant numbers), A302991, A330899.
Sequence in context: A231547 A290141 A087245 * A153501 A215012 A181595
Adjacent sequences: A342097 A342098 A342099 * A342101 A342102 A342103


KEYWORD

nonn


AUTHOR

Jon E. Schoenfield, Feb 27 2021


STATUS

approved



