This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A088012 #62 Jul 28 2025 20:18:17
%S A088012 1155,8925,32445,442365,159030135,815634435,2586415095,
%T A088012 1956860570050575,221753180448460815,747406020889133775
%N A088012 Odd solutions to abs(sigma(k) - 2k) <= log(k). Numbers k whose abundance-radius does not exceed log(k).
%C A088012 This sequence should include odd perfect numbers too, if they exist.
%C A088012 From _Walter Nissen_, Dec 15 2005: (Start)
%C A088012 abundancy(k) k 2k sigma(k) abundance
%C A088012 1.99480519480519 1155 2310 2304 -6
%C A088012 2.00067226890756 8925 17850 17856 6
%C A088012 2.00018492834027 32445 64890 64896 6
%C A088012 2.00001356346004 442365 884730 884736 6
%C A088012 2.00000011318610 159030135 318060270 318060288 18
%C A088012 1.99999999264376 815634435 1631268870 1631268864 -6
%C A088012 2.00000000695943 2586415095 5172830190 5172830208 18
%C A088012 As it happens, abundance of these is -6, 6 or 18. This is not necessarily true for larger terms. (End)
%C A088012 See also A171929 and A188597 and A188263 for sequences of numbers (any / deficient / abundant) whose relative abundancy tends to 2. - _M. F. Hasler_, Feb 19 2017
%C A088012 3278298202600507814120339275775985 is also a term with abundance 30. In fact, it and 815634435 are the only odd terms known where abs(sigma(k)-2k) <= log_10(k). - _Alexander Violette_, Nov 05 2020; updated by _Max Alekseyev_, Jul 27 2025
%C A088012 Also includes 827880257692739174385 and 255286886041240176056063754225. - _Max Alekseyev_, Jul 27 2025
%H A088012 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%e A088012 1155 is in the sequence because sigma(1155) = 2304, giving 2*1155 - 2304 = 6, while natural log of 1155 is about 7.05.
%e A088012 From _M. F. Hasler_, Jul 18 2016: (Start)
%e A088012 We have the following factorizations:
%e A088012 1155 = 3 * 5 * 7 * 11,
%e A088012 8925 = 3 * 5^2 * 7 * 17,
%e A088012 32445 = 3^2 * 5 * 7 * 103,
%e A088012 442365 = 3 * 5 * 7 * 11 * 383,
%e A088012 159030135 = 3^5 * 5 * 11 * 73 * 163,
%e A088012 815634435 = 3 * 5 * 7 * 11 * 547 * 1291,
%e A088012 2586415095 = 3^2 * 5 * 11 * 31 * 41 * 4111.
%e A088012 The sequence appears to be a subsequence of A171929. (End)
%t A088012 abu[x_] := Abs[DivisorSigma[1, x]-2*x] Do[If[ !Greater[abu[n], Log[n]//N]&&OddQ[n], Print[n]], {n, 1, 100000}]
%o A088012 (PARI) is(n)=n%2 && abs(sigma(n)-2*n)<=log(n) \\ _Charles R Greathouse IV_, Feb 21 2017
%Y A088012 Cf. A000079, A000396, A005100, A005101, A077374, A087167, A088007-A088011.
%Y A088012 Cf. also A171929, A188263, A188597, A228059, A295296.
%K A088012 hard,nonn,more
%O A088012 1,1
%A A088012 _Labos Elemer_ and _Farideh Firoozbakht_, Oct 20 2003
%E A088012 a(7) from _Donovan Johnson_, Dec 21 2008
%E A088012 a(9) from _Alexander Violette_ confirmed and a(8), a(10) added by _Max Alekseyev_, Jul 27 2025