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 A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. (Formerly M2226 N0884) 422
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 REFERENCES 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). 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). LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 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]. H. Bottomley, Illustration of initial terms Don Coppersmith, An answer to the problem of Don Saari, 1987. Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers] P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, to appear. Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018. Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014). F. Richman, Aliquot series: Abundant, deficient, perfect Eric Weisstein's World of Mathematics, Restricted Divisor Function Eric Weisstein's World of Mathematics, Divisor Function FORMULA G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006 a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005 a(n) = A155085(-n). - Michael Somos, Sep 20 2011 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) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013 a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013 a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013 Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016 a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018 Erdos (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 EXAMPLE 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 , [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - Omar E. Pol, Nov 24 2019 MAPLE with(numtheory); [ seq(sigma(n)-n, n=1..100) ]; MATHEMATICA Table[ Plus @@ Select[ Divisors[ n ], # 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014 A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015 Cf. A032741, A000203, A048050, A000593, A034090, A034091, A027750. Cf. A051953, A051731. Cf. A037020 (primes), A053868, A053869 (odd and even terms). Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof). Cf. A007956 (products of proper divisors). Cf. A005100, A005101, A000396, A259180, A122726 (related concepts). Sequence in context: A294886 A069250 A294888 * A173455 A324535 A318501 Adjacent sequences:  A001062 A001063 A001064 * A001066 A001067 A001068 KEYWORD nonn,core,easy,nice AUTHOR STATUS approved

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Last modified September 20 23:24 EDT 2020. Contains 337265 sequences. (Running on oeis4.)