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A088012 Odd solutions to abs(sigma(x)-2x) <= log(x). Numbers n whose abundance-radius does not exceed log(n). 17
1155, 8925, 32445, 442365, 159030135, 815634435, 2586415095 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

There are no other terms below 2*10^9.

     Abundancy(n)             n            2n       sigma(n)  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.

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

LINKS

Table of n, a(n) for n=1..7.

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

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

Additional comments from Walter Nissen, Dec 15 2005

a(7) from Donovan Johnson, Dec 21 2008

STATUS

approved

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)