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A033880
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Abundance of n, or (sum of divisors of n) - 2n.
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101
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-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
(list;
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history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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For no known n is a(n) = 1. If there is such an n it must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982). - Jonathan Vos Post, May 01 2011
a(n) = -1 iff n is a power of 2. a(n) = 1 - n 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]
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Since the perfect numbers having density 0, the deficient numbers have density 0.7520 < 1 - A(2) < 0.7526. - Daniel Forgues, Oct 10 2015
2-abundance of n, a special case of the k-abundance of n, defined as (sum of divisors of n) - k*n, k >= 1. - Daniel Forgues, Oct 24 2015
Not to be confused with the abundancy of n, defined as (sum of divisors of n) / n. (Cf. A017665 / A017666.) - Daniel Forgues, Oct 25 2015
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REFERENCES
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Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Section B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.
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LINKS
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J. G. Wurtzel, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file computed by 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
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
Eric Weisstein's World of Mathematics, Abundance
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Quasiperfect Number
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FORMULA
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a(n) = A000203(n) - A005843(n). - Omar E. Pol, Dec 14 2008
a(n) = A001065(n) - n. - Omar E. Pol, Dec 27 2013
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EXAMPLE
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For n = 10 the divisors of 10 are 1, 2, 5, 10. The sum of proper divisors of 10 minus 10 is 1 + 2 + 5 - 10 = -2, so the abundance of 10 is a(10) = -2. - Omar E. Pol, Dec 27 2013
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MAPLE
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with(numtheory); n->sigma(n) - 2*n;
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MATHEMATICA
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Array[Total[Divisors[#]]-2#&, 70] (* Harvey P. Dale, Sep 16 2011 *)
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PROG
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(PARI) a(n)=sigma(n)-2*n \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [SumOfDivisors(n)-2*n: n in [1..100]]; // Vincenzo Librandi, Oct 11 2015
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CROSSREFS
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Equals -A033879.
a(n) = A000203(n) - A005843(n). - Omar E. Pol, Dec 14 2008
a(n) = A001065(n) - n. - Omar E. Pol, Dec 27 2013
Cf. A005100 (a(n) < 0), A000396 (a(n) = 0) and A005101 (a(n) > 0).
See also A023197.
Sequence in context: A233150 A103977 A109883 * A033879 A324546 A033883
Adjacent sequences: A033877 A033878 A033879 * A033881 A033882 A033883
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KEYWORD
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sign,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Definition corrected Jul 04 2005
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STATUS
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approved
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