cp's OEIS Frontend

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

%I A342100 #15 Dec 06 2024 06:49:51
%S A342100 12,18,20,24,40,42,56,60,72,80,84,88,90,102,104,108,112,114,354,366,
%T A342100 368,372,380,384,392,396,400,402,464,468,476,480,492,500,504,552,560,
%U A342100 564,572,576,580,582,650,654,836,840,945,948,952,954,1002,2002,2004,2024
%N A342100 Abundant numbers k at which the ratio (number of abundant numbers in 1..k)/k reaches a new record high.
%C A342100 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...
%C A342100 a(115) = 7254; rho(7254) = 1810/7254 = 0.2495175075820...
%C A342100 Conjecture: a(115) is the final term of this sequence.
%C A342100 This sequence is finite since rho(2212) > A302991 and therefore there is a number N such that abs(rho(n) - A302991) < eps for all n > N and for an arbitrarily small eps > 0. Therefore, the number of values of n for which rho(n) > rho(2212) is finite. - _Amiram Eldar_, Dec 06 2024
%H A342100 Amiram Eldar, <a href="/A342100/b342100.txt">Table of n, a(n) for n = 1..115</a>
%e A342100 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.
%e A342100 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.
%e A342100 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.
%e A342100 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.
%e A342100 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.
%t A342100 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 *)
%Y A342100 Cf. A005101 (abundant numbers), A302991, A330899.
%K A342100 nonn,fini
%O A342100 1,1
%A A342100 _Jon E. Schoenfield_, Feb 27 2021
%E A342100 Keyword "fini" added by _Amiram Eldar_, Dec 06 2024