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%I M3197 N1292 #264 Mar 22 2024 08:47:03
%S 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 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 54,40,72,8,80,30,60,24,62,32,104,1,84,48,68,18,96,48,72,13,74,38,124
%N Sum of odd divisors of n.
%C Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - _Michael Somos_, May 17 2013
%C A069289(n) <= a(n). - _Reinhard Zumkeller_, Apr 05 2015
%C A000203, A001227 and this sequence have the same parity: A053866. - _Omar E. Pol_, May 14 2016
%C 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
%C 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
%C 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].
%C Also, alternating row sums of A126988. - _Omar E. Pol_, Feb 11 2018
%D J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 496, pp. 69-246, Ellipses, Paris, 2004.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.
%D F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000593/b000593.txt">Table of n, a(n) for n = 1..10000</a>
%H Francesca Aicardi, <a href="http://arxiv.org/abs/0806.1273">Matricial formulas for partitions</a>, arXiv:0806.1273 [math.NT], 2008.
%H M. Baake and R. V. Moody, <a href="https://arxiv.org/abs/math/9904028">Similarity submodules and root systems in four dimensions</a>, arXiv:math/9904028 [math.MG], 1999; Canad. J. Math. 51 (1999), 1258-1276.
%H J. A. Ewell, <a href="https://www.fq.math.ca/Papers1/45-3/ewell.pdf">On the sum-of-divisors function</a>, Fib. Q., 45 (2007), 205-207.
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H Heekyoung Hahn, <a href="http://arxiv.org/abs/1507.04426">Convolution sums of some functions on divisors</a>, arXiv:1507.04426 [math.NT], 2015.
%H Kaya Lakein and Anne Larsen, <a href="https://arxiv.org/abs/2107.07637">A Proof of Merca's Conjectures on Sums of Odd Divisor Functions</a>, arXiv:2107.07637 [math.NT], 2021.
%H Mircea Merca, <a href="http://dx.doi.org/10.1007/s11139-016-9856-3">The Lambert series factorization theorem</a>, The Ramanujan Journal, January 2017, also <a href="https://www.researchgate.net/publication/312324402">here</a>
%H Mircea Merca, <a href="https://acad.ro/sectii2002/proceedings/doc2021-2/02-Merca.pdf">Congruence identities involving sums of odd divisors function</a>, Proceedings of the Romanian Academy, Series A, Volume 22, Number 2/2021, pp. 119-125.
%H H. Movasati and Y. Nikdelan, <a href="http://arxiv.org/abs/1603.09411">Gauss-Manin Connection in Disguise: Dwork Family</a>, arXiv:1603.09411 [math.AG], 2016.
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H H. J. Stephen Smith, <a href="https://archive.org/stream/reportofbritisha66brit#page/322/">Report on the Theory of Numbers. — Part VI.</a>, Report of the 35 Meeting of the British Association for the Advancement of Science (1866). See p. 336.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...].
%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
%F a(2*n) = A000203(2*n)-2*A000203(n), a(2*n+1) = A000203(2*n+1). - _Henry Bottomley_, May 16 2000
%F a(2*n) = A054785(2*n) - A000203(2*n). - _Reinhard Zumkeller_, Apr 23 2008
%F Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. - _David W. Wilson_, Aug 01 2001
%F a(n) = Sum_{d divides n} (-1)^(d+1)*n/d. - _Vladeta Jovovic_, Sep 06 2002
%F Sum_{k=1..n} a(k) is asymptotic to c*n^2 where c=Pi^2/24. - _Benoit Cloitre_, Dec 29 2002
%F G.f.: Sum_{n>0} n*x^n/(1+x^n). - _Vladeta Jovovic_, Oct 11 2002
%F G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
%F G.f.: Sum_{k>0} -(-x)^k / (1 - x^k)^2. - _Michael Somos_, Oct 29 2005
%F a(n) = A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). - _Reinhard Zumkeller_, Apr 18 2006
%F From _Joerg Arndt_, Nov 09 2010: (Start)
%F G.f.: Sum_{n>=1} (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)).
%F G.f.: deriv(log(P)) = deriv(P)/P where P = Product_{n>=1} (1 + q^n). (End)
%F Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. Dirichlet convolution of A062157 with A000027. - _R. J. Mathar_, Jun 28 2011
%F a(n) = Sum_{k = 1..A001227(n)} A182469(n,k). - _Reinhard Zumkeller_, May 01 2012
%F 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
%F a(n) = Sum_{k=1..n} k*A000009(k)*A081362(n-k). - _Mircea Merca_, Feb 26 2014
%F a(n) = A000203(n) - A146076(n). - _Omar E. Pol_, Apr 05 2016
%F a(2*n) = a(n). - _Giuseppe Coppoletta_, Nov 02 2016
%F a(n) = n * [x^n] log((-1; x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 21 2016
%F From _Wolfdieter Lang_, Dec 11 2016: (Start)
%F 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.
%F 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.
%F (End)
%F a(n) = A000203(n)/A038712(n). - _Omar E. Pol_, Dec 14 2017
%F a(n) = A000203(n)/(2^(1 + (A183063(n)/A001227(n))) - 1). - _Omar E. Pol_, Nov 06 2018
%F a(n) = A000203(2n) - 2*A000203(n). - _Ridouane Oudra_, Aug 28 2019
%F From _Peter Bala_, Jan 04 2021: (Start)
%F a(n) = (2/3)*A002131(n) + (1/3)*A002129(n) = (2/3)*A002131(n) + (-1)^(n+1)*(1/3)*A113184(n).
%F a(n) = A002131(n) - (1/2)*A146076; a(n) = 2*A002131(n) - A000203(n). (End)
%F a(n) = A000203(A000265(n)) - _John Keith_, Aug 30 2021
%e 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 + ...
%p 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 Table[a := Select[Divisors[n], OddQ[ # ]&]; Sum[a[[i]], {i, 1, Length[a]}], {n, 1, 60}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t f[n_] := Plus @@ Select[ Divisors@ n, OddQ]; Array[f, 75] (* _Robert G. Wilson v_, Jun 19 2011 *)
%t a[ n_] := If[ n < 1, 0, Sum[ -(-1)^d n / d, {d, Divisors[ n]}]]; (* _Michael Somos_, May 17 2013 *)
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# n / # &]]; (* _Michael Somos_, May 17 2013 *)
%t a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d, {d, Divisors[ n]}]]; (* _Michael Somos_, May 17 2013 *)
%t a[ n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Aug 15 2015 *)
%t Array[Total[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] &, {75}] (* _Michael De Vlieger_, Apr 07 2016 *)
%t Table[SeriesCoefficient[n Log[QPochhammer[-1, x]], {x, 0, n}], {n, 1, 75}] (* _Vladimir Reshetnikov_, Nov 21 2016 *)
%t Table[DivisorSum[n,#&,OddQ[#]&],{n,80}] (* _Harvey P. Dale_, Jun 19 2021 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))}; /* _Michael Somos_, May 29 2005 */
%o (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 */
%o (PARI) s=vector(100);for(n=1,100,s[n]=sumdiv(n,d,d*(d%2)));s /* _Zak Seidov_, Sep 24 2011*/
%o (PARI) a(n)=sigma(n>>valuation(n,2)) \\ _Charles R Greathouse IV_, Sep 09 2014
%o (Haskell)
%o a000593 = sum . a182469_row -- _Reinhard Zumkeller_, May 01 2012, Jul 25 2011
%o (Sage) [sum(k for k in divisors(n) if k % 2) for n in (1..75)] # _Giuseppe Coppoletta_, Nov 02 2016
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[j*x^j/(1+x^j): j in [1..2*m]]) )); // _G. C. Greubel_, Nov 07 2018
%o (Magma) [&+[d:d in Divisors(n)|IsOdd(d)]:n in [1..75]]; // _Marius A. Burtea_, Aug 12 2019
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%Y Cf. A000005, A000203, A000265, A001227, A006128, A050999, A051000, A051001, A051002, A078471 (partial sums), A069289, A247837 (subset of the primes).
%Y Cf. A301799, A301800.
%K nonn,core,easy,nice,mult
%O 1,3
%A _N. J. A. Sloane_
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