A342100 Abundant numbers k at which the ratio (number of abundant numbers in 1..k)/k reaches a new record high.

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

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