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A033879 Deficiency of n, or 2n - (sum of divisors of n). 44

%I

%S 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,

%T 0,28,-12,30,1,18,14,22,-19,36,16,22,-10,40,-12,42,4,12,20,46,-28,41,

%U 7,30,6,52,-12,38,-8,34,26,58,-48,60,28,22,1,46,-12,66,10,42,-4,70,-51

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

%C 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

%C a(n) = 1 iff n is a power of 2. a(n) = n - 1 iff n is prime. - _Omar E. Pol_, Jan 30 2014

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

%H T. D. Noe, <a href="/A033879/b033879.txt">Table of n, a(n) for n = 1..2000</a>

%H Nichole Davis, Dominic Klyve and Nicole Kraght, <a href="http://dx.doi.org/10.2140/involve.2013.6.493">On the difference between an integer and the sum of its proper divisors</a>, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493

%H Jose A. B. Dris, <a href="https://arxiv.org/abs/1610.01868">Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers</a>, arXiv preprint arXiv:1610.01868 [math.NT], 2016.

%H Jose Arnaldo B. Dris, <a href="https://arxiv.org/abs/1703.09077">Analysis of the Ratio D(n)/n</a>, arXiv:1703.09077 [math.NT], 2017.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = -A033880(n).

%F a(n) = A005843(n) - A000203(n). - _Omar E. Pol_, Dec 14 2008

%F a(n) = n - A001065(n). - _Omar E. Pol_, Dec 27 2013

%F G.f.: 2*x/(1 - x)^2 - Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 24 2017

%F a(n) = A286385(n) - A252748(n). - _Antti Karttunen_, May 13 2017

%e 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

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

%t Table[2n-DivisorSigma[1,n],{n,80}] (* _Harvey P. Dale_, Oct 24 2011 *)

%o (PARI) a(n)=2*n-sigma(n) \\ _Charles R Greathouse IV_, Oct 13 2016

%Y Cf. A000203, A005101, A033880, A074918, A075728, A192895, A286385, A286449.

%K sign,nice,changed

%O 1,3

%A _N. J. A. Sloane_

%E Definition corrected Jul 04 2005

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Last modified October 21 17:23 EDT 2017. Contains 293698 sequences.