%I A001065 M2226 N0884
%S A001065 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 A001065 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,
%U A001065 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 A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n 
               that are less than n.
%C A001065 Equals row sums of triangle A141846 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 11 2008
%D A001065 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001065 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001065 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 A001065 George E. Andrews, Number Theory. New York: Dover, 1994 . Pages 1, 75-92; 
               p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
%H A001065 T. D. Noe, <a href="b001065.txt">Table of n, a(n) for n = 1..10000</a>
%H A001065 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001065 H. Bottomley, <a href="a1065.gif">Illustration of initial terms</a>
%H A001065 Primefan, <a href="http://primefan.tripod.com/RestrictDivsSum1000.html">
               Sums of Restricted Divisors for n=1 to 1000</a>
%H A001065 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/aliquot.htm">
               Aliquot series:Abundant,deficient,perfect</a>
%H A001065 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RestrictedDivisorFunction.html">Link to a section of The World of 
               Mathematics (1).</a>
%H A001065 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DivisorFunction.html">Link to a section of The World of Mathematics 
               (2).</a>
%H A001065 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001065 G.f.: Sum_{k>0} k x^(2k)/(1-x^k) - Michael Somos Jul 05 2006
%F A001065 a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 02 2005
%F A001065 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
%e A001065 For n=44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
%p A001065 with(numtheory); [ seq(sigma(n)-n,n=1..100) ];
%t A001065 Table[ Plus @@ Select[ Divisors[ n ], #<n & ], {n, 1, 90} ]
%t A001065 Table[Plus @@ Divisors[n] - n, {n, 1, 90}] [From Zak Seidov (zakseidov(AT)yahoo.com), 
               Sep 10 2009]
%o A001065 (PARI) {a(n)=if(n<1, 0, sigma(n)-n)} /* Michael Somos Jul 05 2006 */
%o A001065 (MuPad) numlib::sigma(n)-n$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 13 2008
%Y A001065 Cf. A032741, A000203, A048050, A000593, A034090, A034091, A027750.
%Y A001065 Cf. A051953, A051731, A141846.
%Y A001065 Sequence in context: A134689 A117552 A069250 this_sequence A109646 A145063 
               A007650
%Y A001065 Adjacent sequences: A001062 A001063 A001064 this_sequence A001066 A001067 
               A001068
%K A001065 nonn,core,easy,nice
%O A001065 1,4
%A A001065 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001065 Replaced a geocities.com URL - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 30 2009