A000593 Sum of odd divisors of n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• core
• easy
• nice
• mult

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

a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017

It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].

Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018

Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025

Examples

			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 + ...
		

References

• Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 496, pp. 69-246, Ellipses, Paris, 2004.
• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.
• Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg, 1994, p. 133.
• John 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.
• Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999; Canad. J. Math. 51 (1999), 1258-1276.
• John A. Ewell, On the sum-of-divisors function, Fib. Q., 45 (2007), 205-207.
• 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.
• Kaya Lakein and Anne Larsen, A Proof of Merca's Conjectures on Sums of Odd Divisor Functions, arXiv:2107.07637 [math.NT], 2021.
• Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017, also here
• Mircea Merca, Congruence identities involving sums of odd divisors function, Proceedings of the Romanian Academy, Series A, Volume 22, Number 2/2021, pp. 119-125.
• Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016.
• Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 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.
• Index entries for sequences mentioned by Glaisher.

Crossrefs

Cf. A000005, A000203, A000265, A001227, A006128, A050999, A051000, A051001, A051002, A065442, A078471 (partial sums), A069289, A247837 (subset of the primes).

Cf. A301799, A301800.

Programs

Haskell

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

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[j*x^j/(1+x^j): j in [1..2*m]])  )); // G. C. Greubel, Nov 07 2018

Magma

[&+[d:d in Divisors(n)|IsOdd(d)]:n in [1..75]]; // Marius A. Burtea, Aug 12 2019

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 *)
Table[DivisorSum[n,#&,OddQ[#]&],{n,80}] (* Harvey P. Dale, Jun 19 2021 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))}; /* Michael Somos, May 29 2005 */

PARI

N=66; x='x+O('x^N); Vec( serconvol( log(prod(j=1,N,1+x^j)), 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

Python

from math import prod
from sympy import factorint
def A000593(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) # Chai Wah Wu, Sep 09 2021

Sage

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

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, Dirichlet convolution of A062157 with A000027. - 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

From Joerg Arndt, Nov 09 2010: (Start)

G.f.: Sum_{n>=1} (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)).

G.f.: deriv(log(P)) = deriv(P)/P where P = Product_{n>=1} (1 + q^n). (End)

Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. - R. J. Mathar, Jun 28 2011

a(n) = Sum_{k = 1..A001227(n)} A182469(n,k). - 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), commutating factors of the D.g.f. given above by Jovovic, Oct 11 2002. See also the a(n) version given by Jovovic, Sep 06 2002. (End)

a(n) = A000203(n)/A038712(n). - Omar E. Pol, Dec 14 2017

a(n) = A000203(n)/(2^(1 + (A183063(n)/A001227(n))) - 1). - Omar E. Pol, Nov 06 2018

a(n) = A000203(2n) - 2*A000203(n). - Ridouane Oudra, Aug 28 2019

From Peter Bala, Jan 04 2021: (Start)

a(n) = (2/3)*A002131(n) + (1/3)*A002129(n) = (2/3)*A002131(n) + (-1)^(n+1)*(1/3)*A113184(n).

a(n) = A002131(n) - (1/2)*A146076; a(n) = 2*A002131(n) - A000203(n). (End)

a(n) = A000203(A000265(n)) - John Keith, Aug 30 2021

Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = A065442 - 1 = 0.60669... . - Amiram Eldar, Dec 14 2024