A077374 -id:A077374 - cp's OEIS frontend

A087167 Odd numbers such that sigma(n) - 2n = 6.

8925, 32445, 442365

Offset: 1

Farideh Firoozbakht, Oct 19 2003

If m is in this sequence and 5 doesn't divide m then m is an odd Weird number. There are no other terms up to 2*10^9. Jud McCranie wrote: There are no terms between 2*10^9 and 6.5*10^9.
a(4) > 10^12. - Donovan Johnson, Dec 08 2011
a(4) > 10^13. - Giovanni Resta, Mar 29 2013
a(4) > 10^22. - Wenjie Fang, Jun 16 2014
Any term x of this sequence can be combined with any term y of A141548 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016

			a(1)=8925 because sigma(8925)=2*8925+6 and 8925 is the first odd number such that sigma(n)-2n=6.

R. K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." B2 in Unsolved Problems in Number Theory, 2nd ed.New York:Springer- Verlag, pp. 45-53, 1994.

S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.
C. Rivera, Puzzle 233. A little twist.
Eric Weisstein's World of Mathematics, Weird Number.

Cf. A003380, A077374, A005101, A005835, A141548 (deficiency 6).

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

is(n)=n%2 && sigma(n)==2*n+6 \\ Charles R Greathouse IV, Mar 09 2014

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

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 *)

A088005 Numbers whose abundance is (+-1) times a prime.

Original entry on oeis.org

3, 9, 10, 18, 20, 25, 36, 49, 50, 81, 100, 104, 121, 136, 196, 225, 289, 324, 338, 392, 464, 484, 576, 650, 676, 784, 800, 841, 900, 961, 968, 1058, 1089, 1156, 1225, 1250, 1352, 1682, 1936, 1952, 2209, 2312, 2450, 2500, 2888, 3136, 3364, 3721, 4225, 4356

Offset: 1

Labos Elemer, Oct 18 2003

nonn

			n=18: sigma[18]=18+9+6+3+2+1=39, 2n=36, abundance=3, a prime;
n=25: sigma[25]=25+5+1=31, 2n=50, abundance=-19.

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

Cf. A000203, A000040, A077374, A087998.

ab[x_] := DivisorSigma[1, x]-2*x Do[If[PrimeQ[s=ab[n]], Print[{n, s}]], {n, 1, 10000}]

isok(n) = isprime(abs(sigma(n)-2*n)); \\ Michel Marcus, Sep 21 2019

abs(sigma(n)-2n) is a prime.

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

cp's OEIS Frontend

A087167 Odd numbers such that sigma(n) - 2n = 6.

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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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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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A088005 Numbers whose abundance is (+-1) times a prime.

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