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A001065
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Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
(Formerly M2226 N0884)
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506
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0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106
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OFFSET
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1,4
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COMMENTS
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Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences. Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
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FORMULA
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G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
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EXAMPLE
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x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - Omar E. Pol, Nov 24 2019
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MAPLE
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numtheory[sigma](n)-n ;
end proc:
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MATHEMATICA
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Table[ Plus @@ Select[ Divisors[ n ], #<n & ], {n, 1, 90} ]
Table[Plus @@ Divisors[n] - n, {n, 1, 90}] (* Zak Seidov, Sep 10 2009 *)
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PROG
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(PARI) {a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */
(Haskell)
(Python)
from sympy import divisor_sigma
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CROSSREFS
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Cf. A007956 (products of proper divisors).
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KEYWORD
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nonn,core,easy,nice
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AUTHOR
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STATUS
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approved
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