A188263 -id:A188263 - 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

A188597 Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

Offset: 1

T. D. Noe, Apr 05 2011

nonn

The abundancy of a number k is defined as A(k) = sigma(k)/k. Deficient numbers have an abundancy less than 2. This sequence has terms in common with A171929. Sequence A188263, which deals with abundant numbers, approaches 2 from above. The similar sequence for even numbers consists of the powers of 2.
a(29) > 10^12. - Donovan Johnson, Apr 08 2011
This sequence is finite iff there is an odd perfect number (which would have abundancy 2). Otherwise, one always has a subsequent term a(n+1) <= a(n)*p where p is the smallest prime not dividing a(n) and larger than 1/(2/A(a(n))-1). Indeed, such an a(n)*p is still deficient but has abundancy larger than a(n), thus closer to 2. - M. F. Hasler, Feb 22 2017
From M. F. Hasler, Jan 25 2020: (Start)
The upper bounds a(n)*p mentioned above are often terms of the sequence, but not the subsequent but a later one: e.g., 9*5 = 45, 15*7 = 105, 45*7 = 315, 105*11 = 1155, 315*107 = 33705, 1155*389 = 449295, 26325*389 = 10240425, ...
Is 9 the largest term not divisible by 15? Only 7 of the 26 terms listed after 45 are not multiples of 7: is this subsequence finite? (End)

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

a(22)-a(28) from Donovan Johnson, Apr 08 2011

A335052 Odd unitary abundant numbers whose unitary abundancy is closer to 2 than that of any smaller odd unitary abundant number.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 31395, 33915, 39585, 41055, 45885, 51765, 80535, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 617102535, 724239285, 1756753845

Offset: 1

Amiram Eldar, May 21 2020

nonn

The unitary abundancy of a number k is usigma(k)/k, where usigma(k) is the sum of unitary divisors of k (A034448).

			The unitary abundancies of the first terms are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

Cf. A034448, A129485, A188263, A302570, A335053, A335055.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); seq = {}; r = 3; Do[s = usigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

A335053 Odd bi-unitary abundant numbers whose bi-unitary abundancy is closer to 2 than that of any smaller odd bi-unitary abundant number.

Original entry on oeis.org

945, 25515, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 396165, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 13351635, 132335385, 159030135, 1756753845, 6561644355, 10394173335, 13455037365, 37456183215

Offset: 1

Amiram Eldar, May 21 2020

nonn

The bi-unitary abundancy of a number k is bsigma(k)/k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

			The bi-unitary abundancies of the first terms are 2.031..., 2.005..., 2.0019..., 2.0018..., 2.0015..., ...

Cf. A188999, A188263, A293186, A302571, A335052, A335055.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); seq = {}; r = 3; Do[s = bsigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

A335055 Odd infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller odd infinitary abundant number.

Original entry on oeis.org

945, 29835, 33345, 43065, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 210124665, 719709375, 724239285, 1756753845, 9665740455, 10394173335

Offset: 1

Amiram Eldar, May 21 2020

nonn

The infinitary abundancy of a number k is isigma(k)/k, where isigma(k) is the sum of infinitary divisors of k (A049417).

			The infinitary abundancies of the first terms are 2.031..., 2.027..., 2.015..., 2.006..., 2.001..., ...

Cf. A049417, A127666, A188263, A335052, A335053, A335054.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 3; Do[s = isigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^5, 2}]; seq

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).

All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...

Amiram Eldar, Table of n, a(n) for n = 1..36

The exponential version of A071927.
Subsequence of A001694 and A328136.
Cf. A051377, A057521, A129575, A336253.
Similar sequences: A188263, A302570, A302571, A335054.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; s={}; Do[r = esigma[n]/n; If[r <= 2, Continue[]]; If[r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

a(23)-a(27) from Amiram Eldar, May 06 2025

A336679 Odd exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller odd exponential abundant number.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 985646025, 1150227225, 1566972225, 1685513025, 2105433225, 2679615225, 6485886225, 6554064825, 6933060225, 9150077475, 179678493225, 185601564225, 191620685025, 195686793225

Offset: 1

Amiram Eldar, Jul 30 2020

The exponential abundancy of a number k is esigma(k)/k, where esigma(k) is the sum of exponential divisors of k (A051377).

The corresponding values of esigma(k)/k are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

The exponential version of A188263.
The odd version of A336254.
Subsequence of A321147.
Similar sequences: A335052, A335053, A335055.
Cf. A051377.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; seq = {}; rm = 3; Do[r = esigma[n]/n; If[r > 2 && r < rm, rm = r; AppendTo[seq, n]], {n, 1, 10^9, 2}]; seq

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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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A188597 Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

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A335052 Odd unitary abundant numbers whose unitary abundancy is closer to 2 than that of any smaller odd unitary abundant number.

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A335053 Odd bi-unitary abundant numbers whose bi-unitary abundancy is closer to 2 than that of any smaller odd bi-unitary abundant number.

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A335055 Odd infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller odd infinitary abundant number.

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A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

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