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A033879 Deficiency of n, or 2n - (sum of divisors of n). 63
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

REFERENCES

Jose Arnaldo B. Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, Print ISSN 1310-5132, Online ISSN 2367-8275. Vol. 23, 2017, No. 4, 1-13; http://nntdm.net/papers/nntdm-23/NNTDM-23-4-01-13.pdf

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.

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)

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), A296074, A296075.

Sequence in context: A103977 A109883 A033880 * A033883 A106316 A300721

Adjacent sequences:  A033876 A033877 A033878 * A033880 A033881 A033882

KEYWORD

sign,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition corrected Jul 04 2005

STATUS

approved

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Last modified July 21 15:53 EDT 2018. Contains 312858 sequences. (Running on oeis4.)