A223609 -id:A223609 - 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

A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

Original entry on oeis.org

11, 21, 26, 68, 656, 2336, 8768, 133376, 528896, 34360918016, 35184409837568, 576460757135261696

Offset: 1

Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 15 2004

a(13) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
A subsequence of A274556. a(11) <= b(23) = 35184409837568 ~ 3.5*10^13, since b(k) := 2^(k-1)*(2^k+9) is in this sequence for all k in A057196 (2^k+9 is prime). All known terms except a(2) = 21 are of that form. - M. F. Hasler, Jul 18 2016
Any term x of this sequence can be combined with any term y of A223609 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

			The divisors of 68 are {1, 2, 4, 17, 34, 68} and so sigma(68) = 1 + 2 + 4 + 17+ 24 + 68 = 126 = 2*68 - 10; thus, the deficiency of 68 is 10 so 68 is a term of the sequence.

Cf. A033879, A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).
Cf. also A274556.
Cf. A223609 (abundance 10).

[n: n in [1..9*10^6] | (SumOfDivisors(n)) eq 2*n-10]; // Vincenzo Librandi, Sep 15 2016

Select[ Range[ 85000000], DivisorSigma[1, # ] + 10 == 2# &]

Edited and extended by Robert G. Wilson v, Dec 15 2004
a(10) from Donovan Johnson, Dec 23 2008
Edited by M. F. Hasler, Jul 18 2016
a(11)-a(12) from Hiroaki Yamanouchi, Aug 21 2018

A274554 Numbers k such that sigma(k) == 0 (mod k-4).

Original entry on oeis.org

5, 6, 10, 22, 24, 60, 130, 184, 1012, 2272, 18904, 33664, 70564, 85936, 100804, 391612, 527872, 1090912, 17619844, 2147713024, 6800695312, 34360655872, 549759483904, 1661355408388

Offset: 1

Paolo P. Lava, Jun 28 2016

A125247 is a subset of this sequence.

			sigma(5) (mod 5-4) = 6 (mod 1) = 0.

Cf. A067702, A125247, A223609, A274551, A274552, A274553, A274566.

[n: n in [5..2*10^6] | SumOfDivisors(n) mod (n-4) eq 0 ]; // Vincenzo Librandi, Jul 02 2016

q:= k-> is(irem(numtheory[sigma](k), k-4)=0):
select(q, [$5..400000])[];  # Alois P. Heinz, Jun 14 2025

k = -4; Select[Range[Abs@ k + 1, 10^7], Mod[DivisorSigma[1, #], # + k] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

a(19)-a(24) from Giovanni Resta, Jul 01 2016

A274556 Numbers k such that sigma(k) == 0 (mod k-5).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 11, 12, 18, 21, 26, 68, 656, 2336, 8768, 133376, 528896, 34360918016, 35184409837568, 576460757135261696

Offset: 1

Paolo P. Lava, Jun 30 2016

			sigma(6) (mod 6-5) = 12 mod 1 = 0.

Contains subsequence A101223.

Cf. A067702, A223609, A274551, A274552, A274553, A274554, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274565, A274566.

[n: n in [1..2*10^6] | n ne 5 and SumOfDivisors(n) mod (n-5) eq 0]; // Vincenzo Librandi, Jul 02 2016

n = -5; Select[Range[1, 10^6], # + n != 0 && Mod[DivisorSigma[1, #], # + n] == 0 &] (* Michael De Vlieger, Jul 01 2016 *)

a(18) from Giovanni Resta, Jul 01 2016

Terms 2,3,4 inserted, a(19)-a(20) added by Max Alekseyev, May 25 2025

A274565 Numbers k such that sigma(k) == 0 (mod k+10).

Original entry on oeis.org

14, 176, 1376, 3230, 3770, 6848, 114256, 125696, 544310, 561824, 740870, 2075648, 4199030, 4607296, 8436950, 33468416, 134045696, 199272950, 624032630, 1113445430, 1550860550, 85905593344, 2199001235456, 35184284008448

Offset: 1

Paolo P. Lava, Jul 06 2016

			sigma(14) mod (14 + 10) = 24 mod 24 = 0.

Contains subsequence A223611.
Contains 2 * odd terms of A223609.
Cf. A045770, A067702, A088833, A181598, A274551, A274552, A274553, A274554, A274556, A274557, A274558, A274559, A274560, A274561, A274562, A274563, A274564, A274566.

[n: n in [1..2*10^6] | SumOfDivisors(n) mod (n+10) eq 0 ]; // Vincenzo Librandi, Jul 06 2016

k = 10; Select[Range[Abs@k+1, 10^6], Mod[DivisorSigma[1, #], # + k] == 0 &] (* Vincenzo Librandi, Jul 06 2016 *)

a(13)-a(23) from Giovanni Resta, Jul 06 2016

a(24) from Max Alekseyev, May 29 2025

A275701 Numbers n whose abundance is 26: sigma(n) - 2n = 26.

Original entry on oeis.org

80, 1184, 6464, 29312, 78975, 510464, 557192, 137431875584, 549741658112, 8796036399104, 35184258842624, 2251798907715584

Offset: 1

Timothy L. Tiffin, Aug 05 2016

Any term x = a(m) can be combined with any term y = A275702(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have produced only one amicable pair: (x,y) = (1184,1210) = (a(2),A275702(5)) = (A063990(3),A063990(4)). If more are ever found, then they will also exhibit y-x = 26.
Notice that:
a(1) =     80 =   5* 16 = (2*4^2-27)*(4^2)
a(2) =   1184 =  37* 32 =   (4^3-27)*(4^3)/2
a(3) =   6464 = 101* 64 = (2*4^3-27)*(4^3)
a(4) =  29312 = 229*128 =   (4^4-27)*(4^4)/2
a(6) = 510464 = 997*512 =   (4^5-27)*(4^5)/2.
If p = 2*4^k-27 is prime and n = p*(p+27)/2, then it is not hard to show that sigma(n) - 2*n = 26. The values of k in A275767 will guarantee that p is prime (A275749). Similarly, if q = 4^k-27 is prime and n = q*(q+27)/2, then sigma(n) - 2*n = 26. The values of k in A274519 will guarantee that q is prime (A275750). So, the following values will be in this sequence and provide upper bounds for the next eight terms:
(2*4^9-27)*(4^9)     = 137431875584 >= a(8)
  (4^10-27)*(4^10)/2 = 549741658112 >= a(9)
  (4^11-27)*(4^11)/2 = 8796036399104 >= a(10)
(2*4^11-27)*(4^11)   = 35184258842624 >= a(11)
  (4^13-27)*(4^13)/2 = 2251798907715584 >= a(12)
  (4^25-27)*(4^25)/2 = 633825300114099501099609227264 >= a(13)
  (4^28-27)*(4^28)/2 = 2596148429267412841487728652582912 >= a(14)
  (4^29-27)*(4^29)/2 = 41538374868278617137133892585652224 >= a(15).
a(8) > 10^9. - Michel Marcus, Sep 15 2016
a(8) > 2*10^9. - Michel Marcus, Dec 31 2016
a(13) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

			a(1) = 80, since sigma(80)-2*80 = 186-160 = 26.
a(2) = 1184, since sigma(1184)-2*1184 = 2394-2368 = 26.
a(3) = 6464, since sigma(6464)-2*6464 = 12954-12928 = 26.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A033880, A063990, A274519, A275702 (deficiency 26), A275749, A275750, A275767.

Cf. A223609 (abundance 10), ..., A223613 (abundance 24).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 26]; // Vincenzo Librandi, Sep 16 2016

Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* Vincenzo Librandi, Sep 16 2016 *)

isok(n) = sigma(n) - 2*n == 26; \\ Michel Marcus, Sep 15 2016

a(8)-a(12) from Hiroaki Yamanouchi, Aug 23 2018

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Original entry on oeis.org

134, 284, 410, 632, 1292, 1628, 4064, 9752, 12224, 22712, 66992, 72944, 403988, 556544, 2161664, 2330528, 8517632, 13228352, 14563832, 15422912, 20732792, 89472632, 134733824, 150511232, 283551872, 537903104, 731670272, 915473696, 1846850576, 2149548032, 2159587616

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) in this sequence can be used with any term y in A275996 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.
The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - Max Alekseyev, Aug 27 2025

			a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..89

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A002025, A063990.

Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)

isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

a(23)-a(31) from Jinyuan Wang, Mar 02 2020

A292626 Numbers k whose abundance is 128: sigma(k) - 2*k = 128.

Original entry on oeis.org

860, 5336, 6536, 9656, 16256, 55796, 70864, 98048, 361556, 776096, 2227616, 4145216, 4498136, 4632896, 8124416, 13086016, 34869056, 38546576, 150094976, 172960856, 196066256, 962085536, 1080008576, 1733780336, 1844788112, 2143256576, 2531343872, 2986104064, 9677743616, 11276687456, 17104503968, 20680182272, 21568135616

Offset: 1

Fabian Schneider, Sep 20 2017

Max Alekseyev, Table of n, a(n) for n = 1..87

Subsequence of A259174.
Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64).

fQ[n_] := DivisorSigma[1, n] == 2 n + 128; Select[ Range@ 10^8, fQ] (* Robert G. Wilson v, Nov 19 2017 *)

isok(n) = sigma(n) - 2*n == 128; \\ Michel Marcus, Sep 20 2017

a(9)-a(18) from Michel Marcus, Sep 20 2017
a(19)-a(24), a(26), a(29)-a(30), a(33) from Robert G. Wilson v, Nov 20 2017
Missing terms a(25), a(27)-a(28), a(31)-a(32) inserted and terms a(34) onward added by Max Alekseyev, Aug 30 2025

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984

Offset: 1

Max Alekseyev, Jul 29 2025

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A387352 Numbers m with deficiency 32: sigma(m) - 2*m = -32.

Original entry on oeis.org

250, 376, 1276, 12616, 20536, 396916, 801376, 1297312, 8452096, 33721216, 40575616, 59376256, 89397016, 99523456, 101556016, 150441856, 173706136, 269096704, 283417216, 500101936, 1082640256, 1846506832, 15531546112, 34675557856, 136310177392, 136783784608

Offset: 1

Max Alekseyev, Aug 27 2025

Contains numbers 2^(k-1)*(2^k + 31) for k in A247952.

Max Alekseyev, Table of n, a(n) for n = 1..54

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A275997 (k=64).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A247952.

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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A101223 Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

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Magma

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A274554 Numbers k such that sigma(k) == 0 (mod k-4).

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Magma

Maple

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A274556 Numbers k such that sigma(k) == 0 (mod k-5).

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A274565 Numbers k such that sigma(k) == 0 (mod k+10).

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Magma

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A275701 Numbers n whose abundance is 26: sigma(n) - 2n = 26.

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A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

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