A141545 -id:A141545 - cp's OEIS frontend

A033879 Deficiency of n, or 2n - (sum of divisors of n).

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, -4, 12, 4, 6, 1, 16, -3, 18, -2, 10, 8, 22, -12, 19, 10, 14, 0, 28, -12, 30, 1, 18, 14, 22, -19, 36, 16, 22, -10, 40, -12, 42, 4, 12, 20, 46, -28, 41, 7, 30, 6, 52, -12, 38, -8, 34, 26, 58, -48, 60, 28, 22, 1, 46, -12, 66, 10, 42, -4, 70, -51

Offset: 1

N. J. A. Sloane

Records for the sequence of the absolute values are in A075728 and the indices of these records in A074918. - R. J. Mathar, Mar 02 2007
a(n) = 1 iff n is a power of 2. a(n) = n - 1 iff n is prime. - Omar E. Pol, Jan 30 2014
If a(n) = 1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019
It is not known whether there are any -1's in this sequence. See comment in A033880. - Antti Karttunen, Feb 02 2020

			For n = 10 the divisors of 10 are 1, 2, 5, 10, so the deficiency of 10 is 10 minus the sum of its proper divisors or simply 10 - 5 - 2 - 1 = 2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000 [First 2000 terms from T. D. Noe, terms up to 16384 from Antti Karttunen]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo B. Dris, Analysis of the Ratio D(n)/n, arXiv:1703.09077 [math.NT], 2017.
Jose Arnaldo Bebita Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(4) (2017), 1-13.
Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego, Some Modular Considerations Regarding Odd Perfect Numbers, arXiv:2002.12139 [math.NT], 2020.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, Conditions equivalent to the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers - Part II, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 62-67.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences related to sigma(n).

Cf. A000396 (positions of zeros), A005100 (of positive terms), A005101 (of negative terms).
Cf. A000203, A033880, A074918, A075728, A192895, A286385, A286449, A295881, A295882.
Cf. A083254 (Möbius transform), A228058, A296074, A296075, A323910, A325636, A325826, A325970, A325976.
Cf. A141545 (positions of a(n) = -12).
For this sequence applied to various permutations of natural numbers and some other sequences, see A323174, A323244, A324055, A324185, A324546, A324574, A324575, A324654, A325379.

with(numtheory): A033879:=n->2*n-sigma(n): seq(A033879(n), n=1..100);

Table[2n-DivisorSigma[1,n],{n,80}] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=2*n-sigma(n) \\ Charles R Greathouse IV, Oct 13 2016

from sympy import divisor_sigma
def A033879(n): return (n<<1)-divisor_sigma(n) # Chai Wah Wu, Apr 13 2024

[2*n-sigma(n, 1) for n in range(1, 73)] # Stefano Spezia, Jul 18 2025

a(n) = -A033880(n).
a(n) = A005843(n) - A000203(n). - Omar E. Pol, Dec 14 2008
a(n) = n - A001065(n). - Omar E. Pol, Dec 27 2013
G.f.: 2*x/(1 - x)^2 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 24 2017
a(n) = A286385(n) - A252748(n). - Antti Karttunen, May 13 2017
From Antti Karttunen, Dec 29 2017: (Start)
a(n) = Sum_{d|n} A083254(d).
a(n) = Sum_{d|n} A008683(n/d)*A296075(d).
a(n) = A065620(A295881(n)) = A117966(A295882(n)).
a(n) = A294898(n) + A000120(n).
(End)
From Antti Karttunen, Jun 03 2019: (Start)
Sequence can be represented in arbitrarily many ways as a difference of the form (n - f(n)) - (g(n) - n), where f and g are any two sequences whose sum f(n)+g(n) = sigma(n). Here are few examples:
a(n) = A325314(n) - A325313(n) = A325814(n) - A034460(n) = A325978(n) - A325977(n).
a(n) = A325976(n) - A325826(n) = A325959(n) - A325969(n) = A003958(n) - A324044(n).
a(n) = A326049(n) - A326050(n) = A326055(n) - A326054(n) = A326044(n) - A326045(n).
a(n) = A326058(n) - A326059(n) = A326068(n) - A326067(n).
a(n) = A326128(n) - A326127(n) = A066503(n) - A326143(n).
a(n) = A318878(n) - A318879(n).
a(A228058(n)) = A325379(n). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Pi^2/12 = 0.177532... . - Amiram Eldar, Dec 07 2023

Definition corrected by N. J. A. Sloane, Jul 04 2005

A076496 Numbers k such that sigma(k) == 12 (mod k).

Original entry on oeis.org

1, 6, 11, 24, 30, 42, 54, 66, 78, 102, 114, 121, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 780, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338

Offset: 1

Labos Elemer, Oct 21 2002

nonn

			6*p is a solution if p > 3 is prime, since sigma(6*p) = 1 + 2 + 3 + 6 + p + 2*p + 3*p + 6*p = 12*(p+1) = 2*6*p + 12 = 2*k + 12. These are "regular" solutions. Also k = 121, 304 are "singular" solutions. See other remainders in cross-references.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000, corrected by _Sidney Cadot_, Feb 05 2023

Cf. A054024, A045768, A045769, A045770, A076495, A088834.

Cf. A141545 (a subsequence).

Select[Range[2000], Mod[DivisorSigma[1, #] - 12, #] == 0 &] (* Vincenzo Librandi, Mar 11 2014, corrected by Amiram Eldar, Jan 04 2023 *)

isok(k) = Mod(sigma(k), k) == 12; \\ Michel Marcus, Jan 04 2023

Initial term 1 added by Vincenzo Librandi, Mar 11 2014

Terms 6 and 11 inserted by Michel Marcus, Jan 04 2023

A141549 Numbers k whose deficiency is 12: 2k - sigma(k) = 12.

Original entry on oeis.org

13, 45, 76, 688, 8896, 133888, 537051136, 35184418226176, 144115191028645888, 2305843021024854016

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

Numbers n whose abundance is -12. No other terms up to n=100,000,000. - Jason G. Wurtzel, Aug 24 2010
For all k in A102633, the number 2^(k-1)*(2^k+11) is in this sequence. So far all terms except a(2) are of this form. For k = 55, 71, this yields terms 649037107316853651724695645454336, 2787593149816327892704951291908936712585216. - M. F. Hasler, Apr 23 2015; edited by Max Alekseyev, May 27 2025
Any term x = a(m) can be combined with any term y = A141545(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 not produced an amicable pair. However, if one is ever found, then it will exhibit x-y = 12. - Timothy L. Tiffin, Sep 13 2016
a(11) > 10^20. - Max Alekseyev, May 27 2025

			a(1) = 13, since 2*13 - sigma(13) = 26 - 14 = 12. - _Timothy L. Tiffin_, Sep 13 2016

Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20); A141545 (abundance 12).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -12]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n+12,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 12 &] (* Vincenzo Librandi, Sep 14 2016 *)

for(n=1, 10^8, if(((sigma(n)-2*n)==-12), print1(n, ", "))) \\ Jason G. Wurtzel, Aug 24 2010

a(7) from Donovan Johnson, Dec 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Aug 21 2018
a(10) from Max Alekseyev, May 27 2025

A175989 Numbers with abundance 32.

Original entry on oeis.org

572, 992, 7544, 10184, 28544, 83312, 113072, 122624, 382772, 507392, 537248, 698528, 791264, 1081568, 1279136, 2154584, 2279072, 5029184, 15126992, 29581424, 74899952, 89245784, 95327216, 307801856, 623799776, 712023296, 903230984, 1734487184, 9210347984

Offset: 1

R. J. Mathar, Nov 04 2010

nonn

a(74) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..73 (first 47 terms from Giovanni Resta)

Cf. A005101, A141545-A141547, A181601.

Select[Range[10000000],DivisorSigma[1,#]-2#==32&] (* Harvey P. Dale, Dec 10 2010 *)

is(n)=sigma(n)==2*n+32 \\ Charles R Greathouse IV, Feb 21 2017

{n: A033880(n) = 32}.

Additional terms provided by Harvey P. Dale, Dec 10 2010

a(19)-a(29) from Donovan Johnson, Dec 08 2011

A141546 Numbers whose abundance is 14.

Original entry on oeis.org

272, 7232, 30848, 516608, 134094848, 2146992128, 35184309174272

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

a(7) > 10^12. - Donovan Johnson, Dec 08 2011
a(7) > 10^13. - Giovanni Resta, Mar 29 2013
a(8) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A141550 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
Every number of the form 2^(j-1)*(2^j - 15), where 2^j - 15 is prime (see A059612), is a term. - Jon E. Schoenfield, Jun 02 2019

			a(1) = 272, since sigma(272) - 2*272 = 558 - 544 = 14. - _Timothy L. Tiffin_, Sep 13 2016

Cf. A141550 (deficiency 14), A141545 (abundance 12), A141547 (abundance 16).

[n: n in [1..10^8] | SumOfDivisors(n)- 2*n eq 14]; // Vincenzo Librandi, Mar 20 2015

lst={};Do[If[n==Plus@@Divisors[n]-n-14,AppendTo[lst,n]],{n,10^4}];Print[lst];
lst = {}; Do[ If[2 n + 14 == DivisorSigma[1, n], AppendTo[lst, n]], {n, 2 10^8, 2}]; lst (* Robert G. Wilson v, Aug 17 2008 *)

isok(n) = sigma(n) - 2*n == 14; \\ Michel Marcus, Mar 20 2015

{k: A033880(k) = 14}. - R. J. Mathar, Jun 06 2024

a(5)-a(6) from Donovan Johnson, Dec 21 2008

a(7) 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

A371920 Abundant numbers whose abundance is also an abundant number.

Original entry on oeis.org

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 112, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 176, 180, 186, 198, 204, 208, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 304, 306, 308, 312, 318, 330, 336, 342, 348, 354, 360

Offset: 1

Amiram Eldar, Apr 12 2024

First differs from A125639 at n = 12.
Numbers k such that A033880(k) > 0 and A033880(A033880(k)) > 0.
This sequence is infinite: if m is divisible by 6 and coprime to 5, then 5*m is a term.
All the multiply-perfect numbers (A007691) that are not 1 or perfect (A000396), i.e., the terms of A166069, are terms of this sequence.

			24 is a term since A033880(24) = 12 > 0 and A033880(12) = 4 > 0.

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

Cf. A033880 (abundance), A000396, A007691, A125639.
Subsequence of A005101.
Subsequences: A141545, A166069, A223610, A223611, A223613.

ab[n_] := DivisorSigma[1, n] - 2*n; q[n_] := Module[{k = ab[n]}, k > 0 && ab[k] > 0]; Select[Range[360], q]

ab(n) = sigma(n) - 2*n;
is(n) = {my(k = ab(n)); k > 0 && ab(k) > 0;}

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

A033879 Deficiency of n, or 2n - (sum of divisors of n).

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A076496 Numbers k such that sigma(k) == 12 (mod k).

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A141549 Numbers k whose deficiency is 12: 2k - sigma(k) = 12.

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A175989 Numbers with abundance 32.

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A141546 Numbers whose abundance is 14.

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

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

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