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Search: id:A001065
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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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+0
86
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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
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Equals row sums of triangle A141846 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 11 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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).
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LINKS
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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
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series:Abundant,deficient,perfect
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).
Index entries for "core" sequences
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FORMULA
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G.f.: Sum_{k>0} k x^(2k)/(1-x^k) - Michael Somos Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 02 2005
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 (qntmpkt(AT)yahoo.com), Jul 11 2008
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EXAMPLE
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For n=44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
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MAPLE
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with(numtheory); [ seq(sigma(n)-n, n=1..100) ];
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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}] [From Zak Seidov (zakseidov(AT)yahoo.com), Sep 10 2009]
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PROGRAM
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(PARI) {a(n)=if(n<1, 0, sigma(n)-n)} /* Michael Somos Jul 05 2006 */
(MuPad) numlib::sigma(n)-n$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008
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CROSSREFS
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Cf. A032741, A000203, A048050, A000593, A034090, A034091, A027750.
Cf. A051953, A051731, A141846.
Sequence in context: A134689 A117552 A069250 this_sequence A109646 A145063 A007650
Adjacent sequences: A001062 A001063 A001064 this_sequence A001066 A001067 A001068
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KEYWORD
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nonn,core,easy,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
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Replaced a geocities.com URL - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009
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