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%I M2226 N0884 #275 Mar 10 2024 11:10:02
%S 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,
%T 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,
%U 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
%N Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
%C 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
%C 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
%C 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
%C 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
%C Fixed points are in A000396. - _Alois P. Heinz_, Mar 10 2024
%D 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.
%D George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
%D 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].
%D Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001065/b001065.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H Henry Bottomley, <a href="/A001065/a001065.gif">Illustration of initial terms</a>
%H K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, <a href="http://arxiv.org/abs/2110.14136">Numerical and statistical analysis of aliquot sequences</a>, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
%H Don Coppersmith, <a href="/A001065/a001065.pdf">An answer to the problem of Don Saari</a>, 1987.
%H Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="http://math.dartmouth.edu/~carlp/iterate.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
%H Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
%H Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%H P. Pollack and C. Pomerance, <a href="https://doi.org/10.1090/btran/10">Some problems of Erdős on the sum-of-divisors function</a>, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; <a href="http://pollack.uga.edu/reversal-errata.pdf">errata</a>.
%H Carl Pomerance and Hee-Sung Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper7.pdf">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Math. Comp., to appear (2014).
%H Primefan, <a href="http://primefan.tripod.com/RestrictDivsSum1000.html">Sums of Restricted Divisors for n=1 to 1000</a>
%H F. Richman, <a href="http://math.fau.edu/Richman/mla/aliquot.htm">Aliquot series: Abundant, deficient, perfect</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RestrictedDivisorFunction.html">Restricted Divisor Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - _Michael Somos_, Jul 05 2006
%F a(n) = sigma(n) - n = A000203(n) - n. - _Lekraj Beedassy_, Jun 02 2005
%F a(n) = A155085(-n). - _Michael Somos_, Sep 20 2011
%F 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
%F a(n) = A006128(n) - A220477(n) - n. - _Omar E. Pol_ Jan 17 2013
%F a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - _Wesley Ivan Hurt_, Oct 25 2013
%F Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - _Ilya Gutkovskiy_, Aug 07 2016
%F a(n) = 1 + A048050(n), n > 1. - _R. J. Mathar_, Mar 13 2018
%F 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
%F 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
%F 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
%F a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - _Wesley Ivan Hurt_, Jan 28 2021
%F a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - _José de Jesús Camacho Medina_, Feb 23 2021
%e 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 + ...
%e For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
%e Related concepts: (Start)
%e From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
%e Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
%e Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
%e Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
%e Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
%e 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
%p A001065 := proc(n)
%p numtheory[sigma](n)-n ;
%p end proc:
%p seq( A001065(n),n=1..100) ;
%t Table[ Plus @@ Select[ Divisors[ n ], #<n & ], {n, 1, 90} ]
%t Table[Plus @@ Divisors[n] - n, {n, 1, 90}] (* _Zak Seidov_, Sep 10 2009 *)
%t Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* _Jean-François Alcover_, Apr 25 2013 *)
%t Array[Plus @@ Most@ Divisors@# &, 80] (* _Robert G. Wilson v_, Dec 24 2017 *)
%o (PARI) {a(n) = if( n==0, 0, sigma(n) - n)} /* _Michael Somos_, Sep 20 2011 */
%o (MuPAD) numlib::sigma(n)-n$ n=1..81 // _Zerinvary Lajos_, May 13 2008
%o (Haskell)
%o a001065 n = a000203 n - n -- _Reinhard Zumkeller_, Sep 15 2011
%o (Magma) [SumOfDivisors(n)-n: n in [1..100]]; // _Vincenzo Librandi_, May 06 2015
%o (Python)
%o from sympy import divisor_sigma
%o def A001065(n): return divisor_sigma(n)-n # _Chai Wah Wu_, Nov 04 2022
%Y Least inverse: A070015, A359132.
%Y Values taken: A078923, values not taken: A005114.
%Y Records: A034090, A034091.
%Y First differences: A053246, partial sums: A153485.
%Y a(n) = n - A033879(n) = n + A033880(n). - _Omar E. Pol_, Dec 30 2013
%Y Row sums of A141846 and of A176891. - _Gary W. Adamson_, May 02 2010
%Y Row sums of A176079. - _Mats Granvik_, May 20 2012
%Y Alternating row sums of A231347. - _Omar E. Pol_, Jan 02 2014
%Y a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - _Reinhard Zumkeller_, Apr 05 2013
%Y For n > 1: a(n) = A240698(n,A000005(n)-1). - _Reinhard Zumkeller_, Apr 10 2014
%Y A134675(n) = A007434(n) + a(n). - Conjectured by _John Mason_ and proved by _Max Alekseyev_, Jan 07 2015
%Y Cf. A032741, A000203, A048050, A000593, A027750.
%Y Cf. A051953, A051731.
%Y Cf. A037020 (primes), A053868, A053869 (odd and even terms).
%Y Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
%Y Cf. A007956 (products of proper divisors).
%Y Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).
%K nonn,core,easy,nice
%O 1,4
%A _N. J. A. Sloane_, _R. K. Guy_
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