A188597 -id:A188597 - cp's OEIS frontend

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

1, 3, 5, 7, 9, 11, 15, 21, 315, 1155, 8925, 32445, 442365, 815634435

Offset: 1

Jason Earls, Nov 30 2002

Apart from {1, 3, 5, 7, 9, 11, 15, 21, 315}, subset of A088012. Probably finite. - Charles R Greathouse IV, Mar 28 2011
a(15) > 10^13. - Giovanni Resta, Mar 29 2013
The abundance of the given terms a(1..14) is: (-1, -2, -4, -6, -5, -10, -6, -10, -6, -6, 6, 6, 6, -6). See also A171929, A188263 and A188597 for numbers with abundancy sigma(n)/n close to 2. - M. F. Hasler, Feb 21 2017
a(15) > 10^22. - Wenjie Fang, Jul 13 2017

			sigma(32445) = 64896 and 32445*2 = 64890, which makes the odd number 32445 six away from perfection: A(32445) = 6 and hence in this sequence.

Eric Weisstein's World of Mathematics, Abundance.

Contains odd terms of: A191363, A088831, A125246, A088832, A141548, A087167, A125247, A088833, A101223, A223609.

Cf. A033879, A033880, A000203, A088012.

Select[Range[1, 10^6, 2], -10 <= DivisorSigma[1, #] - 2 # <= 10 &] (* Michael De Vlieger, Feb 22 2017 *)

forstep(n=1,442365,2,if(abs(sigma(n)-2*n)<=10,print1(n,",")))

a(14) from Farideh Firoozbakht, Jan 12 2004

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

Original entry on oeis.org

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

Offset: 1

Labos Elemer and Farideh Firoozbakht, Oct 20 2003

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

			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)

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

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

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

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

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

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

A171929 Odd numbers whose abundancy is closer to 2 than any smaller odd number.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 1155, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 13800465, 14571585, 16286445, 20355825, 20487159, 78524145, 132701205, 159030135, 815634435, 2586415095, 29169504045, 40833636525, 125208115065

Offset: 1

Sergio Pimentel, Jan 05 2010

The (relative) abundancy of n is sigma(n)/n, not sigma(n) - 2n. - M. F. Hasler, Apr 12 2015 [As far as I know, "abundancy" has only this meaning; the much less useful sigma(n) - 2n is called "abundance". - Charles R Greathouse IV, Feb 19 2017]
So far all known perfect numbers (abundancy = 2) are even, cf. A000396 = (6, 28, 496, 8128, ...). It has been conjectured but not proved that there are no odd perfect numbers. This sequence provides the list of odd numbers that approach perfection (odd numbers which abundancy is closer to two than the abundancy of any smaller odd number).
Odd numbers n such that abs(sigma(n)/n-2) < abs(sigma(m)/m-2) for all m < n. That is, each n is closer to being an odd perfect number than the preceding n. Interestingly, if abs(sigma(n)/n-2) is expressed as a reduced fraction, the numerator of the fraction is 2 for 25 out of the first 30 terms. Terms a(29) and a(30) are 127595519865 and 154063853475. - T. D. Noe, Jan 28 2010
Indices of successive minima in the sequence |A000203(n)/n - 2| for odd n. The sequence would terminate at the smallest odd perfect number (if it exists). - Max Alekseyev, Jan 26 2010
This sequence is finite if and only there is an odd perfect number. "If" is evident. "Only if" follows because for any real number r > 1 there is an odd number m relatively prime to a given integer such that 1 < sigma(m)/m < r. For example, take a large enough prime. - Charles R Greathouse IV, Dec 13 2016, corrected Feb 19 2017
Of the initial 40 terms, only term 45 is in A228058 (and also in A228059). - Antti Karttunen, Jan 04 2025

			Example: a(8) = 1155 since sigma(1155)/1155 = 1.9948 which is closer to 2 than any smaller a(n).

Giovanni Resta, Table of n, a(n) for n = 1..40 (terms < 10^13; first 36 terms from T. D. Noe)
Mathworld, Abundancy.
Index entries for sequences where odd perfect numbers must occur, if they exist at all.

Cf. A000203, A000396 (perfect numbers), A053624, A119239, A088012, A117349; A188263 and A188597 (the same but restricted to only abundant resp. deficient numbers).

Cf. also A088012, A228058, A228059.

minDiff=Infinity; k=-1; Table[k=k+2; While[abun=DivisorSigma[1,k]/k; Abs[2-abun] > minDiff, k=k+2]; minDiff=Abs[2-abun]; k, {15}] (* T. D. Noe, Jan 28 2010 *)

m=2; forstep(n=1,10^10,2, t=abs(sigma(n)/n - 2); if(tMax Alekseyev, Jan 26 2010

Name improved by T. D. Noe, Jan 28 2010

More terms from Max Alekseyev, T. D. Noe and J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

Original entry on oeis.org

945, 2205, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 14571585, 20355825, 20487159, 78524145, 159030135, 1756753845, 2586415095, 82014476355, 93128205975, 125208115065, 127595519865, 154063853475, 394247024535, 948907364895

Offset: 1

T. D. Noe, Mar 30 2011

nonn

The abundancy of a number n is defined as sigma(n)/n. Abundant numbers have an abundancy greater than 2. All these numbers must be odd primitive abundant numbers, A006038.
These numbers might be considered the opposite of A119239, which has odd numbers whose abundancy increases. This sequence has terms in common with A171929. A similar sequence for deficient numbers is A188597.
These are odd numbers that are barely abundant. See A071927 for the even version.
a(24) > 10^12. - Donovan Johnson, May 05 2012

Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number)

k = 1; minDiff = 1; Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; abun - 2 > minDiff || abun < 2, k = k + 2]; minDiff = abun - 2; k, {10}]

a(15)-a(16) from Donovan Johnson, Mar 31 2011
a(17)-a(22) from Donovan Johnson, Apr 02 2011
a(23) from Donovan Johnson, May 05 2012

A228450 Deficient numbers with increasing abundancy without being powers of 2.

Original entry on oeis.org

3, 9, 10, 44, 110, 136, 592, 884, 2144, 8384, 18632, 32896, 116624, 391612, 527872, 1090912, 2102272, 8394752, 15370304, 73995392, 536920064, 815634435, 2147516416, 34360131584, 217898810368, 546409576448, 549759483904

Offset: 1

Michel Marcus, Oct 27 2013

Without the additional condition one would have obtained A000079, see "least deficient" comment there. Subsequence of A005100.

			First term is 3 with sigma(n)/n = 4/3 ~ 1.33, then 4 with 13/9 ~ 1.44, then 10 with 9/5 = 1.80.

Cf. A000079, A000203, A005100, A071927, A171929, A188597.

abun[n_] := DivisorSigma[1, n]/n; mx = 0; t = {}; Do[m = abun[n]; If[m < 2 && m > mx && ! IntegerQ[Log[2, n]], mx = m; AppendTo[t, n]], {n, 10000}]; t (* T. D. Noe, Apr 09 2014 *)

lista(nn) = {rab = 0; for (n=1, nn, if (n != 2^valuation(n, 2), ab = sigma(n)/n; if ((ab < 2) && (ab > rab), print1(n, ", "); rab = ab;);););} \\ Michel Marcus, Oct 27 2013

a(21)-a(22) from Michel Marcus, Oct 28 2013

a(23)-a(27) from Donovan Johnson, Nov 13 2013

A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 391612, 15370304, 17619844, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056, 42783299288, 80999455688, 137438691328, 217898810368, 546409576448, 1081071376208, 1661355408388

Offset: 1

Antti Karttunen, Jun 13 2019

nonn

Non-abundant numbers whose deficiency (A033879) is less than their binary weight (A000120).

No other terms below < 2^31.

			815634435 = 3*5*7*11*547*1291 is included as in base-2 (A007088) it is written as 110000100111011001100000000011_2, thus A000120(815634435) = 12, while its nonnegative deficiency (A033879) is 2*815634435 - sigma(815634435) = 6 < 12.

Cf. A000120, A000203, A000396 (subsequence), A005187, A033879, A294898, A295296 (deficiency equals binary weight), A326131, A326132.
Intersection of A263837 and A326133.
Cf. also A087485, A141548, A188597.

isA326138(n) = { my(s=sigma(n)); (s>A005187(n)&&s<=(n+n)); };

a(16)-a(26) from Giovanni Resta, Jun 16 2019

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A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

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

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A171929 Odd numbers whose abundancy is closer to 2 than any smaller odd number.

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A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

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A228450 Deficient numbers with increasing abundancy without being powers of 2.

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A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

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