A087167 -id:A087167 - cp's OEIS frontend

A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

Offset: 1

N. J. A. Sloane

OProject@Home in subproject Weird Engine calculates and stores the weird numbers.
There are no odd weird numbers < 10^17. - Robert A. Hearn (rah(AT)ai.mit.edu), May 25 2005
From Alois P. Heinz, Oct 30 2009: (Start)
The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:
  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215
  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215. (End)
There are no odd weird numbers < 1.8*10^19. - Wenjie Fang, Sep 04 2013
S. Benkowski and P. Erdős (1974) proved that the asymptotic density W of weird numbers is positive. It can be shown that W < 0.0101 (see A005835). - Jaycob Coleman, Oct 26 2013
No odd weird number exists below 10^21. This search was done on the volunteer computing project yoyo@home. - Wenjie Fang, Feb 23 2014
No odd weird number with abundance less than 10^14 exists below 10^28. See Odd Weird Search link. - Wenjie Fang, Feb 25 2015
A weird number k multiplied by a prime p > sigma(k) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor. Sequence A065235 lists odd numbers that can be written in only one way as sum of their divisors, and A122036 lists those which are not in A136446, i.e., not sum of proper divisors > 1. - M. F. Hasler, Jul 30 2016

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 70, p. 24, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 4901 terms from Lukasz Swierczewski)
Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.
S. Benkoski, Are All Weird Numbers Even?, Problem E2308, Amer. Math. Monthly, 79 (7) (1972), 774.
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.
David Eppstein, Eqyptian Fractions.
Wenjie Fang, Searching on the boundary of abundance for odd weird numbers, arXiv:2207.12906 [math.NT], 2022.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
H. J. Hindin, Quasipractical numbers, IEEE Communications Magazine, March 1980, pp. 41-45.
Odd Weird Search, Report on the recently completed batch, Feb 23 2015.
OProject, Weird numbers list.
J. Sandor and B. Crstici, Handbook of number theory II, chapter 1.8. [Broken link]
Eric Weisstein's World of Mathematics, Weird Number.
Wikipedia, Weird number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1992.
Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993.

Cf. A002975, A005101, A005835, A005100, A138850, A087167.

Cf. A210455.

a006037 n = a006037_list !! (n-1)
a006037_list = filter ((== 0) . a210455) a005101_list
-- Reinhard Zumkeller, Jan 21 2013

isA006037 := proc(n)
    isA005101(n) and not isA005835(n) ;
end proc:
for n from 1 do
    if isA006037(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Jun 18 2015

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) fQ[n_] := Block[{d, l, t, i}, If[ DivisorSigma[1, n] > 2n && Mod[n, 6] != 0, d = Take[Divisors[n], {1, -2}]; l = 2^Length[d]; t = Table[ NthSubset[j, d], {j, l - 1}]; i = 1; While[i < l && Plus @@ t[[i]] != n, i++ ]]; If[i == l, True, False]]; Select[ Range[ 20000], fQ[ # ] &] (* Robert G. Wilson v, May 20 2005 *)

is_A006037(n,d=divisors(n),s=vecsum(d)-n,m=#d-1)={ m||return; while(d[m]>n, s-=d[m]; m--); d[m]n, is_A006037(n-d[m], d, s-d[m], m-1) && is_A006037(n, d, s-d[m], m-1), sM. F. Hasler, Mar 30 2008; improved and updated to current PARI syntax by M. F. Hasler, Jul 15 2016

is_A006037(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n,d,s)} \\ Equivalent but slightly faster than the self-contained version above.-- For efficiency, ensure that the argument is even or add "!bittest(n,0) && ..." to check this first. - M. F. Hasler, Jul 17 2016

t=0; A006037=vector(100,i, until( is_A006037(t+=2),); t) \\ M. F. Hasler, Mar 30 2008

More terms from Jud McCranie, Oct 21 2001

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

Original entry on oeis.org

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

A141548 Numbers n whose deficiency is 6.

Original entry on oeis.org

7, 15, 52, 315, 592, 1155, 2102272, 815634435

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

a(9) > 10^12. - Donovan Johnson, Dec 08 2011
a(9) > 10^13. - Giovanni Resta, Mar 29 2013
a(9) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
For all k in A059242, the number m = 2^(k-1)*(2^k+5) is in this sequence. This yields further terms 2^46*(2^47+5), 2^52*(2^53+5), 2^140*(2^141+5), ... All even terms known so far and the initial 7 = 2^0*(2^1+5) are of this form. All odd terms beyond a(2) are of the form a(n) = a(k)*p*q, k < n. We have proved that there is no further term of this form with the a(k) given so far. - M. F. Hasler, Apr 23 2015
A term n of this sequence multiplied by a prime p not dividing it is abundant if and only if p < sigma(n)/6 = n/3-1. For the even terms 592 and 2102272, there is such a prime near this limit (191 resp. 693571) such that n*p is a primitive weird number, cf. A002975. For a(3)=52, the largest such prime, 11, is already too small. Odd weird numbers do not exist within these limits. - M. F. Hasler, Jul 19 2016
Any term x of this sequence can be combined with any term y of A087167 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) = 7, since 2*7 - sigma(7) = 14 - 8 = 6. - _Timothy L. Tiffin_, Sep 13 2016

Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.

Cf. A087485 (odd terms).
Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20).
Cf. A087167 (abundance 6).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -6]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n+6,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 6 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n-6 \\ Charles R Greathouse IV, Apr 23 2015, corrected by M. F. Hasler, Jul 18 2016

a(8) from Donovan Johnson, Dec 08 2011

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

A274551 Numbers k such that sigma(k) == 0 (mod k+3).

Original entry on oeis.org

4, 8925, 32445, 442365

Offset: 1

Paolo P. Lava, Jun 28 2016

a(5) > 10^8 if it exists. - Felix Fröhlich, Jul 01 2016
No more terms < 6.5*10^14. - Jud McCranie, Dec 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(4) mod (4+3) = 7 mod 7 = 0.

Cf. A000203, A067702, A087167, A088834, A274552, A274553, A274554, A274556.

[n: n in [1..2*10^6] | SumOfDivisors(n) mod (n+3) eq 0 ]; // Vincenzo Librandi, Jul 02 2016

Select[Range[10^6], Mod[DivisorSigma[1, #], # + 3] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

is(n) = Mod(sigma(n), n+3)==0 \\ Felix Fröhlich, Jul 01 2016

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

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

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

A117350 Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

Original entry on oeis.org

70, 110, 120, 650, 672, 884, 1155, 4030, 5830, 8925, 11096, 17816, 18632, 18904, 30240, 32445, 32760, 45356, 70564, 77744, 85936, 91388, 100804, 116624, 244036, 254012, 388076, 391612, 430272, 442365, 523776, 1090912, 1848964, 2178540

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.
The 2-perfect numbers are excluded because they are 2^n * prime.

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

Donovan Johnson, Table of n, a(n) for n = 1..114 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117346 through A117349.

Offset corrected by Donovan Johnson, Oct 01 2012

cp's OEIS Frontend

A006037 Weird numbers: abundant (A005101) but not pseudoperfect (A005835).

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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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A141548 Numbers n whose deficiency is 6.

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Magma

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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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A274551 Numbers k such that sigma(k) == 0 (mod k+3).

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Magma

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

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

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