A074918 -id:A074918 - 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

A075728 Records in abs(sigma(n)-2*n) (absolute value of A033879).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 120, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 186, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 264, 268, 270, 276, 280, 282, 292, 306

Offset: 1

N. J. A. Sloane, Oct 03 2002

nonn

RECORDS transform of |A033879|.

R. J. Mathar and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 220 terms from R. J. Mathar)
N. J. A. Sloane, Transforms

Cf. A033879, A074918. Different from A006093.

lista(nn) = {rec = -1; for (n=1, nn, d = abs(sigma(n) - 2*n); if (d > rec, print1(d, ", "); rec = d;););} \\ Michel Marcus, Nov 02 2013

Corrected name, Michel Marcus, Nov 02 2013

A053140 Odd primes that are not highly-imperfect numbers.

Original entry on oeis.org

181, 241, 251, 257, 263, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653

Offset: 1

Joseph L. Pe, Oct 03 2002

nonn

The sequence of highly-imperfect numbers begins with 1 and the odd primes up to 113... then even numbers appear and some odd primes are left out.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A074918.

R:= NULL: count:= 0: m:= 0: q:= 1:
for k from 1 do
  v:= abs(numtheory:-sigma(k)-2*k);
  if v > m then
    m:= v;
    V:= select(isprime,[seq(i,i=q .. k-1,2)]);
    count:= count + nops(V);
    R:= R, op(V);
    q:= k + 1 + (k mod 2);
    if count > 100 then break fi
  fi
od:
R; # Robert Israel, Jun 19 2025

More terms from Sean A. Irvine, Dec 11 2021

A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma(k)-k|, where sigma(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

1, 2, 7, 10, 13, 17, 22, 27, 28, 32, 38, 45, 52, 60, 63, 67, 77, 95, 105, 130, 137, 143, 157, 158, 175, 193, 203, 247, 297, 315, 357, 423, 462, 472, 473, 578, 675, 682, 742, 770, 787, 1012, 1138, 1215, 1417, 1463, 1732, 1957, 2047, 2048, 2327, 2363, 2632

Offset: 1

Paolo P. Lava, Jan 04 2012

nonn

Anti-imperfect numbers are anti-deficient numbers or anti-abundant numbers.

			n=1. Anti-divisors: 0. |0-1|=1
n=2. Anti-divisors: 0. |0-2|=2
n=3. Anti-divisors: 2. |2-3|=1 less than 2: 3 is not in the sequence.
n=4. Anti-divisors: 3. |3-4|=1 less than 2: 4 is not in the sequence.
n=5. Anti-divisors: 2,3. |5-3|=2 equal to the maximum: 5 is not in the sequence.
n=6. Anti-divisors: 4. |4-6|=2 equal to the maximum: 6 is not in the sequence.
n=7. Anti-divisors: 2,3,5. |10-7|=3 new maximum: 7 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..600 (terms 1..100 from Paolo P. Lava)

Cf. A066417, A074918, A192267, A192268.

P:=proc(i)
local a,k,n,s;
s:=0;
for n from 1 to i do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)- k/2)<1 then a:=a+k; fi; od;
  if abs(n-a)>s then s:=abs(n-a); print(n); fi;
od;
end:
P(3000);

sig[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]]; d[n] := Abs[sig[n] - n]; s = {}; dm = -1; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 2700}]; s (* Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

cp's OEIS Frontend

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

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A075728 Records in abs(sigma(n)-2*n) (absolute value of A033879).

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A053140 Odd primes that are not highly-imperfect numbers.

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A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma*(k)-k|, where sigma*(k) is the sum of the anti-divisors of k.

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A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma(k)-k|, where sigma(k) is the sum of the anti-divisors of k.