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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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126
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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;
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history;
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internal format)
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OFFSET
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1,4
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COMMENTS
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Row sums of A141846. - Gary W. Adamson, Jul 11 2008
Row sums of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik May 20 2012
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
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
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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).
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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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, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences
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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
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
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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.
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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, Sep 10 2009]
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
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PROG
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(PARI) {a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */
(MuPad) numlib::sigma(n)-n$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008
(Haskell)
a001065 n = a000203 n - n -- Reinhard Zumkeller, Sep 15 2011
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CROSSREFS
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Cf. A032741, A000203, A048050, A000593, A034090, A034091, A027750.
Cf. A051953, A051731, A141846.
Cf. A176891 [From Gary W. Adamson, May 02 2010]
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Sequence in context: A117552 A069250 * A173455 A168111 A109646 A199783
Adjacent sequences: A001062 A001063 A001064 * A001066 A001067 A001068
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KEYWORD
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nonn,core,easy,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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EXTENSIONS
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Replaced a geocities.com URL - R. J. Mathar, Oct 30 2009
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STATUS
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approved
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