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A000593
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Sum of odd divisors of n.
(Formerly M3197 N1292)
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48
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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, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, 57, 31, 72, 14, 54, 40, 72, 8, 80, 30, 60, 24, 62, 32, 104, 1, 84, 48, 68, 18, 96, 48, 72, 13, 74, 38, 124
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OFFSET
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1,3
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COMMENTS
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Denoted by Delta(n) in Glaisher 1907. - Michael Somos, May 17 2013
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REFERENCES
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J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 8).
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
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
Francesca Aicardi, Matricial formulae for partitions, arXiv:0806.1273
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Odd Divisor Function
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for "core" sequences
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FORMULA
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Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...]
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
a(2*n)=A000203(2*n)-2*A000203(n), a(2*n+1)=A000203(2*n+1) - Henry Bottomley, May 16 2000
a(2*n) = A054785(2*n) - A000203(n). - Reinhard Zumkeller, Apr 23 2008
Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001.
a(n) = Sum_{d divides n} (-1)^(d+1)*n/d. - Vladeta Jovovic, Sep 06 2002
Sum(k=1, n, a(k)) is asymptotic to c*n^2 where c=Pi^2/24. - Benoit Cloitre, Dec 29, 2002
G.f.: Sum_{n>0} n*x^n/(1+x^n). - Vladeta Jovovic, Oct 11 2002
G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
G.f.: Sum_{k>0} -(-x)^k/(1-x^k)^2. - Michael Somos, Oct 29 2005
a(n)=A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). - Reinhard Zumkeller, Apr 18 2006
G.f.: sum(n=1, infinity, (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)) ). G.f.: deriv(log(P))=deriv(P)/P where P=prod(n>=1, 1+q^n). [Joerg Arndt, Nov 09 2010]
Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. Dirichlet convolution of A062157 with A000027. - R. J. Mathar, Jun 28 2011
a(n) = sum(A182469(n,k): k = 1..A001227(n)). [Reinhard Zumkeller, May 01 2012]
G.f.: -1/Q(0), where Q(k)= (x-1)*(1-x^(2*k+1)) + x*(-1 +x^(k+1))^4/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
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EXAMPLE
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x + x^2 + 4*x^3 + x^4 + 6*x^5 + 4*x^6 + 8*x^7 + x^8 + 13*x^9 + 6*x^10 + 12*x^11 + ...
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MAPLE
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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;
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MATHEMATICA
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Table[a := Select[Divisors[n], OddQ[ # ]&]; Sum[a[[i]], {i, 1, Length[a]}], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
f[n_] := Plus @@ Select[ Divisors@ n, OddQ]; Array[f, 75] (* Robert G. Wilson v, June 19 2011 *)
a[ n_] := If[ n < 1, 0, Sum[ -(-1)^d n / d, {d, Divisors[ n]}]] (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# n / # &]] (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d, {d, Divisors[ n]}]]
(* Michael Somos, May 17 2013 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))} /* Michael Somos, May 29 2005 */
(PARI) x='x+O('x^66); Vec( serconvol( log(prod(j=1, N, 1+x^j)), Ser(sum(j=1, N, j*x^j)) )) /* Joerg Arndt, May 03 2008, edited by M. F. Hasler, Jun 19 2011 */
(PARI) s=vector(100); for(n=1, 100, s[n]=sumdiv(n, d, d*(d%2))); s /* Zak Seidov, Sep 24 2011*/
(Haskell)
a000593 = sum . a182469_row -- Reinhard Zumkeller, May 01 2012, Jul 25 2011
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CROSSREFS
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Cf. A000005, A000203, A001227, A050999, A051000, A051001, A051002, A078471.
Sequence in context: A192066 A098986 * A115607 A076717 A200360 A120422
Adjacent sequences: A000590 A000591 A000592 * A000594 A000595 A000596
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KEYWORD
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nonn,core,easy,nice,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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