OFFSET
1,1
COMMENTS
This sequence should include odd perfect numbers too, if they exist.
There are no other terms below 2*10^9.
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)
a(8) > 10^13. - Giovanni Resta, Mar 31 2013
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
From Alexander Violette, Nov 05 2020: (Start)
a(8) <= 221753180448460815.
a(9) <= 3278298202600507814120339275775985.
221753180448460815 and 3278298202600507814120339275775985 are also terms of this sequence and their abundances are -30 and 30 respectively. In fact, 3278298202600507814120339275775985 and 815634435 are the only odd terms known where abs(sigma(k)-2k) <= log_10(k). (End)
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
KEYWORD
hard,nonn,more
AUTHOR
Labos Elemer and Farideh Firoozbakht, Oct 20 2003
EXTENSIONS
a(7) from Donovan Johnson, Dec 21 2008
STATUS
approved