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A000593 Sum of odd divisors of n.
(Formerly M3197 N1292)
89
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013

A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015

A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016

For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016

REFERENCES

H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.

Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3; https://www.researchgate.net/publication/312324402_The_Lambert_series_factorization_theorem

H Movasati, Y Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411, 2016

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Francesca Aicardi, Matricial formulas for partitions, arXiv:0806.1273 [math.NT], 2008.

M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

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. 4 and p. 8).

Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.

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

H. J. Stephen Smith, Report on the Theory of Numbers. — Part VI., Report of the 35 Meeting of the British Association for the Advancement of Science (1866). See p. 336.

Eric Weisstein's World of Mathematics, Odd Divisor Function

Eric Weisstein's World of Mathematics, Partition Function Q

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Index entries for "core" sequences

FORMULA

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(2*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

a(n) = sum_{k=1..n} k*A000009(k)*A081362(n-k). - Mircea Merca, Feb 26 2014

a(n) = A000203(n) - A146076(n). - Omar E. Pol, Apr 05 2016

a(2*n) = a(n). - Giuseppe Coppoletta, Nov 02 2016

a(n) = n * [x^n] log((-1; x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 21 2016

From Wolfdieter Lang, Dec 11 2016 : (Start)

G.f.: Sum_{n>=1} x^n*(1+x^(2*n))/(1-x^2*n)^2, from the second to last equation of the proof to Theorem 382 (with x^2 -> x) of the Hardy-Wright reference, p. 312.

a(n) = Sum_{d|n} (-d)*(-1)^(n/d), from the g.f. given above by Jovovic, Oct 11 2002. See also the a(n) version given by Jovovic, Sep 06 2002.

(End)

EXAMPLE

G.f. = 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 + ...

MAPLE

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;

MATHEMATICA

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, Jun 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 *)

a[ n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 15 2015 *)

Array[Total[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] &, {75}] (* Michael De Vlieger, Apr 07 2016 *)

Table[SeriesCoefficient[n Log[QPochhammer[-1, x]], {x, 0, n}], {n, 1, 75}] (* Vladimir Reshetnikov, Nov 21 2016 *)

PROG

(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*/

(PARI) a(n)=sigma(n>>valuation(n, 2)) \\ Charles R Greathouse IV, Sep 09 2014

(Haskell)

a000593 = sum . a182469_row  -- Reinhard Zumkeller, May 01 2012, Jul 25 2011

(Sage) [sum(k for k in divisors(n) if k%2==1) for n in (1..75)] # Giuseppe Coppoletta, Nov 02 2016

CROSSREFS

Cf. A000005, A000203, A001227, A050999, A051000, A051001, A051002, A078471, A069289, A247837 (subset of the primes).

Sequence in context: A192066 A098986 * A115607 A076717 A200360 A228451

Adjacent sequences:  A000590 A000591 A000592 * A000594 A000595 A000596

KEYWORD

nonn,core,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 22 22:05 EDT 2017. Contains 283901 sequences.