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

Offset: 1

Author

Labos Elemer and Farideh Firoozbakht, Oct 20 2003

Keywords

• hard
• nonn
• more

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 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

Also includes 827880257692739174385 and 255286886041240176056063754225. - Max Alekseyev, Jul 27 2025

Examples

			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)
		

Links

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

Crossrefs

Cf. A000079, A000396, A005100, A005101, A077374, A087167, A088007-A088011.

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

Programs

Mathematica

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

PARI

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

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