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A033879 Deficiency of n, or 2n - (sum of divisors of n). 126
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

REFERENCES

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 2000 terms from T. D. Noe)

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, 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.

Index entries for sequences related to sigma(n)

FORMULA

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)

a(n) = A318878(n) - A318879(n) = A000120(n) - (A000203(n)-A005187(n)).

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(A228058(n)) = A325379(n).

(End)

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

For this sequence applied to various permutations of natural numbers and some other injective sequences, see A323174, A323244, A324055, A324185, A324546, A324574, A324575, A324654, A325379.

Sequence in context: A103977 A109883 A033880 * A324546 A033883 A106316

Adjacent sequences:  A033876 A033877 A033878 * A033880 A033881 A033882

KEYWORD

sign,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition corrected Jul 04 2005

STATUS

approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)