A088012 -id:A088012 - 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

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

A088007 Numbers n such that abs(sigma(n) - 2n) <= sqrt(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 16, 18, 20, 28, 32, 44, 50, 52, 64, 70, 88, 104, 110, 128, 130, 136, 152, 174, 184, 186, 196, 222, 232, 246, 256, 258, 272, 282, 304, 315, 318, 354, 366, 368, 402, 426, 438, 464, 474, 496, 498, 512, 534, 550, 582, 592, 606, 618, 642, 650, 654

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Abundance-radius = abs(sigma(n)-2n) does not exceed square root.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A077374, A088008-A088012, A000396, A000079, A005100, A005101.

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

is(n)=abs(sigma(n)-2*n)<=sqrtint(n) \\ Charles R Greathouse IV, Mar 09 2014

New name from Charles R Greathouse IV, Mar 09 2014

A087485 Odd numbers n such that 2n - sigma(n) = 6.

Original entry on oeis.org

7, 15, 315, 1155, 815634435

Offset: 1

Farideh Firoozbakht, Oct 23 2003

This is a subsequence of A077374.
Except for the first term, all known terms of this sequence are divisible by 15. Is there a number n > 1 such that gcd(a(n),3)=1 or gcd(a(n),5)=1?
a(6) > 10^13. - Giovanni Resta, Mar 29 2013
Also, a subsequence of A141548. - M. F. Hasler, Apr 12 2015
The terms a(3) through a(5) are of the form a(k)*p*q, but I have proved that there is no other term of this form with k <= 5. - M. F. Hasler, Apr 13 2015
The terms are also of the form a(n) = 2*p(n) + 1, with primes p(n) = 3, 7, 157, 577, 407817217. All but the last one are such that 2*p(n) - 1 = a(n) - 2 is again prime. - M. F. Hasler, Nov 27 2016
Terms a(2..5) satisfy 2*a(n) - nextprime(sigma(a(n))) = (-1)^n, see also A067795. - M. F. Hasler, Feb 14 2017

			15 is in the sequence because 2*15-sigma(15)=6.

Cf. A077374, A087167, A088011, A088012.

Do[If[OddQ[n]&&2n-DivisorSigma[1, n]==6, Print[n]], {n, 2*10^9}]

is(n)=bittest(n,0)&&sigma(n)+6==2*n \\ M. F. Hasler, Apr 12 2015

a(3) = a(2)*3*7; a(4) = a(2)*7*11 with 7 = precprime(a(2)*2/3), 11=nextprime(a(2)*2/3); a(5) = a(4)*547*1291. - M. F. Hasler, Apr 13 2015

A088011 Even and odd solutions to abs(sigma(x)-2x) <= log(x). Numbers n whose abundance-radius does not exceed log(n).

Original entry on oeis.org

4, 6, 8, 10, 16, 20, 28, 32, 64, 70, 88, 104, 110, 128, 136, 152, 256, 464, 496, 512, 592, 650, 884, 1024, 1155, 1888, 1952, 2048, 2144, 4030, 4096, 5830, 8128, 8192, 8384, 8925, 11096, 16384, 17816, 18632, 18904, 32128, 32445, 32768, 32896, 33664, 45356

Offset: 1

Labos Elemer, Oct 20 2003

nonn

See A088012 for the subsequence of odd terms, only 7 being known up to 10^13. - M. F. Hasler, Feb 21 2017

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A088007-A088012, A077374, A005100, A005101, A000079, A000396.

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

(is(n)=abs(sigma(n)-2*n)M. F. Hasler, Feb 21 2017

A088834 Numbers k such that sigma(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 6, 25, 180, 8925, 32445, 442365

Offset: 1

Labos Elemer, Oct 29 2003

For each integer j in A059609, 2^(j-1)*(2^j - 7) is in the sequence. E.g., for j = A059609(1) = 39 we get 151115727449904501489664. - M. F. Hasler and Farideh Firoozbakht, Dec 03 2013
No more terms to 10^10. - Charles R Greathouse IV, Dec 05 2013
a(9) > 10^13. - Giovanni Resta, Apr 02 2014
a(9) > 1.5*10^14. - Jud McCranie, Jun 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			Sigma(25) = 31 = 1*25 + 6, so 31 mod 25 = 6.

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. Article ID 10.3.1.

Cf. A054024, A045768, A045769, A045770, A077374, A076496, A088012.
Cf. A087167 (a subsequence).
Cf. A059609.

Select[Range[1000000], Mod[DivisorSigma[1, #] - 6, #] == 0 &] (* T. D. Noe, Dec 03 2013 *)

isok(n) = Mod(sigma(n), n) == 6; \\ Michel Marcus, Jan 03 2023

Terms corrected by Charles R Greathouse IV and Farideh Firoozbakht, Dec 03 2013

A088008 Solutions to sigma(x) - 2x <= x^(1/3), both even and odd. Abundance-radius = abs(sigma(n)-2n) does not exceed 3rd root of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 16, 20, 28, 32, 64, 70, 88, 104, 110, 128, 136, 152, 256, 315, 464, 496, 512, 592, 650, 836, 884, 1012, 1024, 1155, 1696, 1758, 1842, 1866, 1878, 1888, 1902, 1952, 1986, 2022, 2048, 2082, 2094, 2118, 2144, 2154, 2202, 2238, 2272, 2274, 2298

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A077374, A088007-A088012, A000396, A000079, A005100, A005101.

abu[x_] := Abs[DivisorSigma[1, x]-2*x] Do[If[ !Greater[abu[n], n^(1/3)//N], Print[n]], {n, 1, 100000}]
Select[Range[2300],Abs[DivisorSigma[1,#]-2#]<=CubeRoot[#]&] (* Harvey P. Dale, Feb 22 2023 *)

A088010 Odd numbers n such that abs(sigma(n)-2n) <= n^(1/3). Abundance-radius = abs(sigma(n)-2n) does not exceed cubic root of n and n is odd.

Original entry on oeis.org

1, 315, 1155, 8415, 8925, 31815, 32445, 33705, 34335, 78975, 351351, 430815, 437745, 442365, 449295, 730125, 1805475, 7667625, 13800465, 14571585, 16029405, 16286445, 20297745, 20355825, 20487159, 21003885, 22982505, 23082885

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A077374, A088007-A088012, A000396, A000079, A005100, A005101.

abu[x_] := Abs[DivisorSigma[1, x]-2*x] Do[If[ !Greater[abu[n], n^(1/3)//N]&&OddQ[n], Print[n]], {n, 1, 100000}]

isok(n) = (n % 2) && (abs(sigma(n)-2*n) < sqrtn(n, 3)); \\ Michel Marcus, Nov 10 2017

a(17)-a(28) from Donovan Johnson, Feb 01 2009

A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 70, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 110, 113, 120, 127, 128, 131, 136, 137, 139, 149, 151, 152, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

			70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.

asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1,n],n]},If[p>n/2,p=n-p];p<=Log[n]];
Select[Range[200],asmlQ] (* Harvey P. Dale, Dec 25 2013 *)

First term prepended by Harvey P. Dale, Dec 25 2013

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

Original entry on oeis.org

6, 10, 20, 28, 70, 88, 104, 110, 120, 136, 152, 464, 496, 592, 650, 672, 884, 1155, 1888, 1952, 2144, 4030, 5830, 8128, 8384, 8925, 11096, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32896, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 116624

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Log is the natural logarithm. Sigma is the sum_of_divisors function.

			70 is a term because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

R. K. Guy, Unsolved Problems in Number Theory, B2.

Donovan Johnson, Table of n, a(n) for n = 1..180 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117347, A117348, A117350.

sigma(n) = k*n + r, abs(r) <= log(n).

Offset corrected by Donovan Johnson, Oct 01 2012

cp's OEIS Frontend

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

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

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A088007 Numbers n such that abs(sigma(n) - 2n) <= sqrt(n).

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A087485 Odd numbers n such that 2n - sigma(n) = 6.

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A088011 Even and odd solutions to abs(sigma(x)-2x) <= log(x). Numbers n whose abundance-radius does not exceed log(n).

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A088834 Numbers k such that sigma(k) == 6 (mod k).

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A088008 Solutions to sigma(x) - 2x <= x^(1/3), both even and odd. Abundance-radius = abs(sigma(n)-2n) does not exceed 3rd root of n.

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A088010 Odd numbers n such that abs(sigma(n)-2n) <= n^(1/3). Abundance-radius = abs(sigma(n)-2n) does not exceed cubic root of n and n is odd.

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