A088012 Odd solutions to abs(sigma(k) - 2k) <= log(k). Numbers k whose abundance-radius does not exceed log(k).

1155, 8925, 32445, 442365, 159030135, 815634435, 2586415095, 1956860570050575, 221753180448460815, 747406020889133775, 827880257692739174385

Offset

1,1

Comments

This sequence should include odd perfect numbers too, if they exist.

From Walter Nissen, Dec 15 2005: (Start)

abundancy(k) k 2k sigma(k) abundance

1.99480519480519 1155 2310 2304 -6

2.00067226890756 8925 17850 17856 6

2.00018492834027 32445 64890 64896 6

2.00001356346004 442365 884730 884736 6

2.00000011318610 159030135 318060270 318060288 18

1.99999999264376 815634435 1631268870 1631268864 -6

2.00000000695943 2586415095 5172830190 5172830208 18

As it happens, abundance of these is -6, 6 or 18. This is not necessarily true for larger terms. (End)

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

3278298202600507814120339275775985 is also a term with abundance 30. In fact, it and 815634435 are two of only three odd terms known where abs(sigma(k)-2k) <= log_10(k). - Alexander Violette, Nov 05 2020; updated by Max Alekseyev, Jul 27 2025

Also includes 255286886041240176056063754225. - Max Alekseyev, Jul 27 2025

Also includes 3133639738039068908629117662878760945, 1960326205542141554690232016958706407178195, and 7644350909445166782402084283654804786139717999288520974394487224777343298764845. - Alexander Violette, Aug 29 2025 and Dec 03 2025

Links

Table of n, a(n) for n=1..11.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Example

1155 is in the sequence because sigma(1155) = 2304, giving 2*1155 - 2304 = 6, while natural log of 1155 is about 7.05.
From M. F. Hasler, Jul 18 2016: (Start)
We have the following factorizations:
1155 = 3 * 5 * 7 * 11,
8925 = 3 * 5^2 * 7 * 17,
32445 = 3^2 * 5 * 7 * 103,
442365 = 3 * 5 * 7 * 11 * 383,
159030135 = 3^5 * 5 * 11 * 73 * 163,
815634435 = 3 * 5 * 7 * 11 * 547 * 1291,
2586415095 = 3^2 * 5 * 11 * 31 * 41 * 4111.
The sequence appears to be a subsequence of A171929. (End)

Mathematica

abu[x_] := Abs[DivisorSigma[1, x]-2*x] Do[If[ !Greater[abu[n], Log[n]//N]&&OddQ[n], Print[n]], {n, 1, 100000}]

Prog

(PARI) is(n)=n%2 && abs(sigma(n)-2*n)<=log(n) \\ Charles R Greathouse IV, Feb 21 2017

Crossrefs

Cf. A000079, A000396, A005100, A005101, A077374, A087167, A088007, A088008, A088010, A088011.

Cf. also A171929, A188263, A188597, A228059, A295296.

Sequence in context: A234498 A147577 A211133 * A046406 A284846 A136355

Adjacent sequences: A088009 A088010 A088011 * A088013 A088014 A088015

Keyword

hard,nonn,more

Author

Labos Elemer and Farideh Firoozbakht, Oct 20 2003

Extensions

a(7) from Donovan Johnson, Dec 21 2008

a(9) from Alexander Violette confirmed and a(8), a(10) added by Max Alekseyev, Jul 27 2025

a(11) from Max Alekseyev, Jan 13 2026

Status

approved