A325970 -id:A325970 - 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

A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

Original entry on oeis.org

2, 4, 3, 8, 4, 2, 4, 16, 12, 9, 6, 14, 6, 12, 8, 32, 10, 26, 8, 21, 14, 18, 12, 20, 30, 16, 18, 2, 14, 24, 10, 64, 16, 24, 22, 88, 14, 30, 26, 36, 18, 32, 14, 42, 26, 28, 24, 54, 56, 80, 20, 32, 26, 40, 36, 60, 38, 42, 30, 56, 18, 42, 48, 128, 42, 48, 22, 50, 28, 72, 26, 122, 26, 54, 58, 46, 48, 56, 26, 86, 120, 60, 42, 96, 54

Offset: 1

Antti Karttunen and David A. Corneth, May 26 2019, with better name from Charlie Neder, Jun 02 2019

nonn

Number of ways to form the sum sigma(n) = x+y so that n-x and n-y are coprime, with x and y in range 0..sigma(n).
From Antti Karttunen, May 28 - Jun 08 2019: (Start)
Empirically, it seems that a(n) >= A034444(n) and also that a(n) >= A034444(A000203(n)) unless n is in A000396.
Specifically, if it could be proved that a(n) >= A034444(n)/2 for n >= 2, which in turn would imply that a(n) >= A001221(n) for all n, then we would know that no odd perfect numbers could exist. Note that a(n) must be 2 on all perfect numbers, whether even or odd. See also A325819.
(End)

			For n=1, sigma(1) = 1, both gcd(1-0, 1-(1-0)) = gcd(1,0) = 1 and gcd(1-1, 1-(1-1)) = gcd(0,1) = 1, thus a(1) = 2.
--
For n=3, sigma(3) = 4, we have 5 cases to consider:
  gcd(3-0, 3-(4-0)) = 1 = gcd(3-4, 3-(4-4)),
  gcd(3-1, 3-(4-1)) = 2 = gcd(3-3, 3-(4-3)),
  gcd(3-2, 3-(4-2)) = 1,
of which three cases give 1 as a result, thus a(3) = 3.
--
For n=6, sigma(6) = 12, we have 13 cases to consider:
  gcd(6-0, 6-(12-0)) = 6 = gcd(6-12, 6-(12-12)),
  gcd(6-1, 6-(12-1)) = 5 = gcd(6-11, 6-(12-11)),
  gcd(6-2, 6-(12-2)) = 4 = gcd(6-10, 6-(12-10)),
  gcd(6-3, 6-(12-3)) = 3 = gcd(6-9, 6-(12-9)),
  gcd(6-4, 6-(12-4)) = 2 = gcd(6-8, 6-(12-8))
  gcd(6-5, 6-(12-5)) = 1 = gcd(6-7, 6-(12-7)),
  gcd(6-6, 6-(12-6)) = 0,
of which only two give 1 as a result, thus a(6) = 2.
--
For n=10, sigma(10) = 18, we have 19 cases to consider:
  gcd(10-0, 10-(18-0)) = 2 = gcd(10-18, 10-(18-18)),
  gcd(10-1, 10-(18-1)) = 1 = gcd(10-17, 10-(18-17)),
  gcd(10-2, 10-(18-2)) = 2 = gcd(10-16, 10-(18-16)),
  gcd(10-3, 10-(18-3)) = 1 = gcd(10-15, 10-(18-15)),
  gcd(10-4, 10-(18-4)) = 2 = gcd(10-14, 10-(18-14)),
  gcd(10-5, 10-(18-5)) = 1 = gcd(10-13, 10-(18-13)),
  gcd(10-6, 10-(18-6)) = 2 = gcd(10-12, 10-(18-12)),
  gcd(10-7, 10-(18-7)) = 1 = gcd(10-11, 10-(18-11)),
  gcd(10-8, 10-(18-8)) = 2 = gcd(10-10, 10-(18-10)),
  gcd(10-9, 10-(18-9)) = 1,
of which 9 cases give 1 as a result, thus a(10) = 9.
--
For n=15, sigma(15) = 24, we have 25 cases to consider:
  gcd(15-0, 15-(24-0)) = 3 = gcd(15-24, 15-(24-24)),
  gcd(15-1, 15-(24-1)) = 2 = gcd(15-23, 15-(24-23)),
  gcd(15-2, 15-(24-2)) = 1 = gcd(15-22, 15-(24-22)),
  gcd(15-3, 15-(24-3)) = 6 = gcd(15-21, 15-(24-21)),
  gcd(15-4, 15-(24-4)) = 1 = gcd(15-20, 15-(24-20)),
  gcd(15-5, 15-(24-5)) = 2 = gcd(15-19, 15-(24-19)),
  gcd(15-6, 15-(24-6)) = 3 = gcd(15-18, 15-(24-18)),
  gcd(15-7, 15-(24-7)) = 2 = gcd(15-17, 15-(24-17)),
  gcd(15-8, 15-(24-8)) = 1 = gcd(15-16, 15-(24-16)),
  gcd(15-9, 15-(24-9)) = 6 = gcd(15-15, 15-(24-15)),
  gcd(15-10, 15-(24-10)) = 1 = gcd(15-14, 15-(24-14)),
  gcd(15-11, 15-(24-11)) = 2 = gcd(15-13, 15-(24-13)),
  gcd(15-12, 15-(24-12)) = 3,
of which 2*4 = 8 cases give 1 as a result, thus a(15) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000010, A000203, A000396, A001221, A014567, A034444, A058062, A062401, A228058, A325807, A325815, A325816, A325817, A325818, A325819, A325960, A325961, A325962, A325965, A325966, A325967, A325968, A325970, A325971, A325972.

Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* Michael De Vlieger, Jun 09 2019 *)

A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };

a(n) = Sum_{i=0..sigma(n)} [1 == gcd(n-i,n-(sigma(n)-i))], where [ ] is the Iverson bracket and sigma(n) is A000203(n).
a(A000396(n)) = 2.
a(n) = A325815(n) + A034444(n).
a(n) = 1+A000203(n) - A325816(n).
a(A228058(n)) = A325819(n).

A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 7, 4, 7, 1, 15, 1, 9, 7, 15, 1, 20, 1, 21, 11, 13, 1, 35, 6, 13, 13, 1, 1, 41, 1, 31, 13, 19, 13, 55, 1, 21, 17, 49, 1, 53, 1, 39, 31, 23, 1, 75, 8, 43, 19, 43, 1, 65, 17, 63, 23, 31, 1, 107, 1, 33, 41, 63, 19, 77, 1, 57, 25, 73, 1, 122, 1, 39, 49, 61, 19, 89, 1, 105, 40, 43, 1, 139, 23, 43, 31, 91, 1, 143, 19, 75

Offset: 1

Antti Karttunen, May 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000040, A000203, A000396, A001065, A033879, A324213, A325817, A325818, A325969, A325970, A325976.

A325826(n) = { my(s=sigma(n)); forstep(k=s-n, 0, -1, if(1==gcd((n+n-sigma(n)), k), return(k))); };

A325818(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(s-i))); };
A325826(n) = (A325818(n) - n);

a(n) = A325818(n) - n = A001065(n) - A325817(n) = A325976(n) - A033879(n).
a(A000040(n)) = a(A000396(n)) = 1.
a(n) >= A325969(n).
gcd(a(n), A325976(n)) = 1.

A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 0, 1, 4, 1, 3, 1, 0, 1, 0, 1, 5, 1, 5, 1, 0, 1, 3, 1, 0, 1, 0, 1, 5, 1, 0, 1, 6, 1, 7, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 1, 3, 1, 5, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 7, 1, 0, 1, 2, 1, 0, 1, 5, 1, 5, 1, 0, 1, 9, 1, 3, 1, 9, 1, 0, 1, 5, 1, 0, 1, 0, 1, 5

Offset: 1

Antti Karttunen, May 29 2019

nonn

By definition, if n is neither an odd prime nor an odd perfect number, then a(n) >= (A020639(n)-1).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A324213, A325817, A325818, A325826, A325961, A325962, A325965, A325966, A325970.

Cf. A006005 (positions of zeros, provided no odd perfect numbers exist).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325960(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i-n))); (0); };

a(n) = (A325961(n) - A325962(n)) / 2, assuming no odd perfect numbers exist.

a(2n) = 1.

A325971 a(n) is the least k >= A007947(n) such that -n + k and (n-sigma(n))+k are relatively prime.

Original entry on oeis.org

1, 2, 4, 2, 6, 7, 8, 2, 3, 11, 12, 7, 14, 15, 16, 2, 18, 7, 20, 11, 22, 23, 24, 7, 5, 27, 4, 27, 30, 31, 32, 2, 34, 35, 36, 6, 38, 39, 40, 11, 42, 43, 44, 23, 16, 47, 48, 7, 7, 10, 52, 27, 54, 7, 56, 15, 58, 59, 60, 31, 62, 63, 22, 2, 66, 67, 68, 35, 70, 71, 72, 7, 74, 75, 16, 39, 78, 79, 80, 11, 3, 83, 84, 43, 86, 87, 88, 23, 90, 31

Offset: 1

Antti Karttunen, May 31 2019

nonn

a(n) is the least k >= A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A007947, A324213, A325817, A325961, A325965, A325970, A325972.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325971(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); (0); };
A325971(n) = { my(s=sigma(n)); for(k=A007947(n), s, if(1==gcd(-n + k, (n-sigma(n))+k), return(k))); };

a(n) = A000203(n) - A325972(n).

a(n) = n - A325970(n).

A325972 a(n) is the largest k <= sigma(n)-A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 5, 0, 13, 10, 7, 0, 21, 0, 9, 8, 29, 0, 32, 0, 31, 10, 13, 0, 53, 26, 15, 36, 29, 0, 41, 0, 61, 14, 19, 12, 85, 0, 21, 16, 79, 0, 53, 0, 61, 62, 25, 0, 117, 50, 83, 20, 71, 0, 113, 16, 105, 22, 31, 0, 137, 0, 33, 82, 125, 18, 77, 0, 91, 26, 73, 0, 188, 0, 39, 108, 101, 18, 89, 0, 175, 118, 43, 0, 181, 22

Offset: 1

Antti Karttunen, May 31 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A007947, A324213, A325818, A325962, A325966, A325970, A325971.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325972(n) = { my(s=sigma(n)); forstep(i=s-A007947(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325971(n).

cp's OEIS Frontend

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

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A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

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A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

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A325971 a(n) is the least k >= A007947(n) such that -n + k and (n-sigma(n))+k are relatively prime.

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A325972 a(n) is the largest k <= sigma(n)-A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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