%I A000593 M3197 N1292
%S A000593 1,1,4,1,6,4,8,1,13,6,12,4,14,8,24,1,18,13,20,6,32,12,24,4,31,14,40,8,
%T A000593 30,24,32,1,48,18,48,13,38,20,56,6,42,32,44,12,78,24,48,4,57,31,72,14,
%U A000593 54,40,72,8,80,30,60,24,62,32,104,1,84,48,68,18,96,48,72,13,74,38,124
%N A000593 Sum of odd divisors of n.
%C A000593 a(2*n) = A054785(2*n) - A000203(n). - from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 23 2008
%D A000593 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000593 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000593 Aicardi Francesca, MATRICIAL FORMULAE FOR PARTITIONS, arXiv:0806.1273.
%D A000593 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
%D A000593 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
%H A000593 T. D. Noe, <a href="b000593.txt">Table of n, a(n) for n = 1..10000</a>
%H A000593 M. Baake and R. V. Moody, <a href="http://arXiv.org/abs/math.MG/9904028">
               Similarity submodules and root systems in four dimensions</a>, Canad. 
               J. Math. 51 (1999), 1258-1276.
%H A000593 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A000593 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A000593 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OddDivisorFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A000593 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionQ.html">Link to a section of The World of Mathematics.</
               a>
%H A000593 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000593 Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...]
%F A000593 Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
%F A000593 a(2n)=A000203(2n)-2*A000203(n), a(2n+1)=A000203(2n+1) - Henry Bottomley 
               (se16(AT)btinternet.com), May 16 2000
%F A000593 Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. 
               - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000593 a(n) = Sum_{d divides n} (-1)^(d+1)*n/d. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 06 2002
%F A000593 Sum(k=1, n, a(k)) is asymptotic to c*n^2 where c=Pi^2/24. - Benoit Cloitre, 
               Dec 29, 2002
%F A000593 G.f.: Sum_{n>0} nx^n/(1+x^n). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 11 2002
%F A000593 G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
%F A000593 G.f.: Sum_{k>0} -(-x)^k/(1-x^k)^2. - Michael Somos Oct 29 2005
%F A000593 a(n)=A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2006
%F A000593 Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)). - Ralf Stephan, Jun 17 
               2007
%p A000593 A000593 := proc(n) local d,s; s := 0; for d from 1 by 2 to n do if n 
               mod d = 0 then s := s+d; fi; od; RETURN(s); end;
%t A000593 Table[a := Select[Divisors[n], OddQ[ # ]&];Sum[a[[i]],{i,1,Length[a]}], 
               {n, 1, 60}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%o A000593 (PARI) a(n)=if(n<1,0,sumdiv(n,d,(-1)^(d+1)*n/d))
%o A000593 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A000593 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A000593 c=sum(j=1,N,j*x^j); \\ log case
%o A000593 s=-log(prod(j=1,N,(1+x^j)^(1))); \\ A000593 Sum of odd divisors of n.
%o A000593 s=serconvol(s,c)
%o A000593 v=Vec(s)
%Y A000593 Cf. A000005, A000203, A001227, A050999-A051002, A078471.
%Y A000593 Sequence in context: A127555 A117001 A098986 this_sequence A115607 A076717 
               A120422
%Y A000593 Adjacent sequences: A000590 A000591 A000592 this_sequence A000594 A000595 
               A000596
%K A000593 nonn,core,easy,nice,mult
%O A000593 1,3
%A A000593 N. J. A. Sloane (njas(AT)research.att.com).