|
%I M2329 N0921 #563 Feb 26 2024 14:46:11
%S 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,31,
%T 42,40,56,30,72,32,63,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U 124,57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,127,84,144,68,126,96,144
%N a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).
%C Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
%C Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
%C A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
%C a(n) is the number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001 [In the language of group theory, a(n) is the number of index-n subgroups of Z x Z. - _Jianing Song_, Nov 05 2022]
%C The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * Product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
%C Sum of number of common divisors of n and m, where m runs from 1 to n. - _Naohiro Nomoto_, Jan 10 2004
%C a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
%C Let s(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + a(n-15) - a(n-22) - a(n-26) + ..., then a(n) = s(n) if n is not pentagonal, i.e., n != (3 j^2 +- j)/2 (cf. A001318), and a(n) is instead s(n) - ((-1)^j)*n if n is pentagonal. - _Gary W. Adamson_, Oct 05 2008 [corrected Apr 27 2012 by _William J. Keith_ based on Ewell and by _Andrey Zabolotskiy_, Apr 08 2022]
%C Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - _Jon Perry_, Nov 08 2012
%C Also total number of parts in the partitions of n into equal parts. - _Omar E. Pol_, Jan 16 2013
%C Note that sigma(3^4) = 11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^(p-1)) = b^p has no solutions b > 2, q prime, p odd prime. - _N. J. A. Sloane_, Dec 21 2013, based on postings to the Number Theory Mailing List by _Vladimir Letsko_ and _Luis H. Gallardo_
%C Limit_{m->infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2 = Pi^2/12 (A072691). See more at A244583. - _Richard R. Forberg_, Jan 04 2015
%C a(n) + A000005(n) is an odd number iff n = 2m^2, m>=1. - _Richard R. Forberg_, Jan 15 2015
%C a(n) = a(n+1) for n = 14, 206, 957, 1334, 1364 (A002961). - _Zak Seidov_, May 03 2016
%C Also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) whose structure arises after the k-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593, where k is an angle greater than zero and less than 180 degrees. - _Omar E. Pol_, Jul 05 2016
%C Equivalent to the Riemann hypothesis: a(n) < H(n) + exp(H(n))*log(H(n)), for all n>1, where H(n) is the n-th harmonic number (Jeffrey Lagarias). See A057641 for more details. - _Ilya Gutkovskiy_, Jul 05 2016
%C a(n) is the total number of even parts in the partitions of 2*n into equal parts. More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts (the comment dated Jan 16 2013 is the case for k = 1). - _Omar E. Pol_, Nov 18 2019
%C From _Jianing Song_, Nov 05 2022: (Start)
%C a(n) is also the number of order-n subgroups of C_n X C_n, where C_n is the cyclic group of order n. Proof: by the correspondence theorem in the group theory, there is a one-to-one correspondence between the order-n subgroups of C_n X C_n = (Z x Z)/(nZ x nZ) and the index-n subgroups of Z x Z containing nZ x nZ. But an index-n normal subgroup of a (multiplicative) group G contains {g^n : n in G} automaticly. The desired result follows from the comment from _Naohiro Nomoto_ above.
%C The number of subgroups of C_n X C_n that are isomorphic to C_n is A001615(n). (End)
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 141, 166.
%D H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
%D Kanold, Hans Joachim, Kreisteilungspolynome und ungerade vollkommene Zahlen. (German), Ber. Math.-Tagung Tübingen 1946, (1947). pp. 84-87.
%D M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
%D A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.1, page 77.
%D G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
%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).
%D Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p. 361.
%H Daniel Forgues, <a href="/A000203/b000203.txt">Table of n, a(n) for n = 1..100000</a> (first 20000 terms from N. J. A. Sloane)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H B. Apostol and L. Petrescu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Apostol/apostol3.html">Extremal Orders of Certain Functions Associated with Regular Integers (mod n)</a>, Journal of Integer Sequences, 2013, # 13.7.5.
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">Quasicrystalline combinatorics</a>
%H H. Bottomley, <a href="/A000203/a000203.gif">Illustration of initial terms</a>
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=SigmaFunction">sigma function</a>
%H Imanuel Chen and Michael Z. Spivey, <a href="http://soundideas.pugetsound.edu/summer_research/238">Integral Generalized Binomial Coefficients of Multiplicative Functions</a>, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
%H D. Christopher and T. Nadu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Christopher/chris7.html">Partitions with Fixed Number of Sizes</a>, Journal of Integer Sequences, 15 (2015), #15.11.5.
%H J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012. - From N. J. A. Sloane, Dec 25 2012
%H Jason Earls, <a href="http://fs.unm.edu/SNJ14.pdf#page=243">The Smarandache sum of composites between factors function</a>, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
%H L. Euler, <a href="http://eulerarchive.maa.org/docs/translations/E175en.pdf">Discovery of a most extraordinary law of numbers, relating to the sum of their divisors</a>
%H L. Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/243/">Observatio de summis divisorum</a>
%H L. Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009.
%H J. A. Ewell, <a href="http://dx.doi.org/10.1090/S0002-9939-1977-0441836-8">Recurrences for the sum of divisors</a>, Proc. Amer. Math. Soc. 64 (2) 1977.
%H F. Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H Johan Gielis and Ilia Tavkhelidze, <a href="https://arxiv.org/abs/1904.01414">The general case of cutting of GML surfaces and bodies</a>, arXiv:1904.01414 [math.GM], 2019.
%H J. W. L. Glaisher, <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=PPN600494829_0020%7CLOG_0017">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
%H J. W. L. Glaisher, <a href="/A002171/a002171.pdf">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
%H M. J. Grady, <a href="http://www.jstor.org/stable/30037550">A group theoretic approach to a famous partition formula</a>, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
%H Masazumi Honda and Takuya Yoda, <a href="https://arxiv.org/abs/2203.17091">String theory, N = 4 SYM and Riemann hypothesis</a>, arXiv:2203.17091 [hep-th], 2022.
%H Douglas E. Iannucci, <a href="https://arxiv.org/abs/1910.11835">On sums of the small divisors of a natural number</a>, arXiv:1910.11835 [math.NT], 2019.
%H Antti Karttunen, <a href="/A000203/a000203.txt">Scheme program for computing this sequence</a>.
%H P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113.
%H M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>, arXiv:math.CO/0503436, 2005.
%H K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources</a>
%H P. Pollack and C. Pomerance, <a href="https://doi.org/10.1090/btran/10">Some problems of Erdős on the sum-of-divisors function</a>, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; <a href="http://pollack.uga.edu/reversal-errata.pdf">errata</a>.
%H Carl Pomerance and Hee-Sung Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper7.pdf">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Math. Comp. 83 (2014), 1903-1913.
%H J. S. Rutherford, <a href="https://doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Act. Cryst. (1992) A48, 500-508. - _N. J. A. Sloane_, Mar 14 2009
%H J. S. Rutherford, <a href="https://doi.org/10.1107/S0108767392007657">The enumeration and symmetry-significant properties of derivative lattices II</a>, Acta Cryst. A49 (1993), 293-300. - _N. J. A. Sloane_, Mar 14 2009
%H John S. Rutherford, <a href="http://dx.doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisor_function">Divisor function</a>
%H <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - _David W. Wilson_, Aug 01 2001
%F For the following bounds and many others, see Mitrinovic et al. - _N. J. A. Sloane_, Oct 02 2017
%F If n is composite, a(n) > n + sqrt(n).
%F a(n) < n*sqrt(n) for all n.
%F a(n) < (6/Pi^2)*n^(3/2) for n > 12.
%F G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n>=1} (1-x^n). - _Joerg Arndt_, Mar 14 2010
%F L.g.f.: -log(Product_{j>=1} (1-x^j)) = Sum_{n>=1} a(n)/n*x^n. - _Joerg Arndt_, Feb 04 2011
%F Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
%F a(n) is odd iff n is a square or twice a square. - _Robert G. Wilson v_, Oct 03 2001
%F a(n) = a(n*prime(n)) - prime(n)*a(n). - _Labos Elemer_, Aug 14 2003 (Clarified by _Omar E. Pol_, Apr 27 2016)
%F a(n) = n*A000041(n) - Sum_{i=1..n-1} a(i)*A000041(n-i). - _Jon Perry_, Sep 11 2003
%F a(n) = -A010815(n)*n - Sum_{k=1..n-1} A010815(k)*a(n-k). - _Reinhard Zumkeller_, Nov 30 2003
%F a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - _Reinhard Zumkeller_, Nov 17 2004
%F Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005
%F G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k / (1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - _Michael Somos_, Apr 05 2003. See the Hardy-Wright reference, p. 312. first equation, and p. 250, Theorem 290. - _Wolfdieter Lang_, Dec 09 2016
%F For odd n, a(n) = A000593(n). For even n, a(n) = A000593(n) + A074400(n/2). - _Jonathan Vos Post_, Mar 26 2006
%F Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - _Gary W. Adamson_, May 20 2007
%F A127093 * [1/1, 1/2, 1/3, ...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, ...]. Row sums of triangle A135539. - _Gary W. Adamson_, Oct 31 2007
%F a(n) = A054785(2*n) - A000593(2*n). - _Reinhard Zumkeller_, Apr 23 2008
%F a(n) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - _Vladimir Kruchinin_, Aug 10 2010
%F Dirichlet convolution of A037213 and A034448. - _R. J. Mathar_, Apr 13 2011
%F G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = -2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2011
%F a(n) = A001065(n) + n. - _Mats Granvik_, May 20 2012
%F a(n) = A006128(n) - A220477(n). - _Omar E. Pol_, Jan 17 2013
%F a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by _Omar E. Pol_, Feb 02 2013, and proved by _Max Alekseyev_, Nov 17 2013
%F a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - _Mircea Merca_, Mar 05 2014
%F a(n) = A240698(n, A000005(n)). - _Reinhard Zumkeller_, Apr 10 2014
%F a(n) = Sum_{d^2|n} A001615(n/d^2) = Sum_{d^3|n} A254981(n/d^3). - _Álvar Ibeas_, Mar 06 2015
%F a(3*n) = A144613(n). a(3*n + 1) = A144614(n). a(3*n + 2) = A144615(n). - _Michael Somos_, Jul 19 2015
%F a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - _Michel Lagneau_, Oct 14 2015
%F a(n) = A000593(n) + A146076(n). - _Omar E. Pol_, Apr 05 2016
%F a(n) = A065475(n) + A048050(n). - _Omar E. Pol_, Nov 28 2016
%F a(n) = (Pi^2*n/6)*Sum_{q>=1} c_q(n)/q^2, with the Ramanujan sums c_q(n) given in A054533 as a c_n(k) table. See the Hardy reference, p. 141, or Hardy-Wright, Theorem 293, p. 251. - _Wolfdieter Lang_, Jan 06 2017
%F G.f. also (1 - E_2(q))/24, with the g.f. E_2 of A006352. See e.g., Hardy, p. 166, eq. (10.5.5). - _Wolfdieter Lang_, Jan 31 2017
%F From _Antti Karttunen_, Nov 25 2017: (Start)
%F a(n) = A048250(n) + A162296(n).
%F a(n) = A092261(n) * A295294(n). [This can be further expanded, see comment in A291750.] (End)
%F a(n) = A000593(n) * A038712(n). - _Ivan N. Ianakiev_ and _Omar E. Pol_, Nov 26 2017
%F a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is the Ramanujan's sum function given in A054533. - _Daniel Suteu_, Jun 14 2018
%F a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - _Daniel Suteu_, Jun 21 2018
%F a(n) = (2^(1 + (A000005(n) - A001227(n))/(A000005(n) - A183063(n))) - 1)*A000593(n) = (2^(1 + (A183063(n)/A001227(n))) - 1)*A000593(n). - _Omar E. Pol_, Nov 03 2018
%F a(n) = Sum_{i=1..n} tau(gcd(n, i)). - _Ridouane Oudra_, Oct 15 2019
%F From _Peter Bala_, Jan 19 2021: (Start)
%F G.f.: A(x) = Sum_{n >= 1} x^(n^2)*(x^n + n*(1 - x^(2*n)))/(1 - x^n)^2 - differentiate equation 5 in Arndt w.r.t. x, and set x = 1.
%F A(x) = F(x) + G(x), where F(x) is the g.f. of A079667 and G(x) is the g.f. of A117004. (End)
%F a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 07 2021
%F With the convention that a(n) = 0 for n <= 0 we have the recurrence a(n) = t(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where t(n) = (-1)^(m+1)*(2*m+1)*n/3 if n = m*(m + 1)/2, with m positive, is a triangular number else t(n) = 0. For example, n = 10 = (4*5)/2 is a triangular number, t(10) = -30, and so a(10) = -30 + 3*a(9) - 5*a(7) + 7*a(4) = -30 + 39 - 40 + 49 = 18. - _Peter Bala_, Apr 06 2022
%F Recurrence: a(p^x) = p*a(p^(x-1)) + 1, if p is prime and for any integer x. E.g., a(5^3) = 5*a(5^2) + 1 = 5*31 + 1 = 156. - _Jules Beauchamp_, Nov 11 2022
%F Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = A319462. - _Vaclav Kotesovec_, May 07 2023
%e For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
%e Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.
%p with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);
%t Table[ DivisorSigma[1, n], {n, 100}]
%t a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* _Michael Somos_, Apr 25 2013 *)
%o (Magma) [SumOfDivisors(n): n in [1..70]];
%o (Magma) [DivisorSigma(1,n): n in [1..70]]; // _Bruno Berselli_, Sep 09 2015
%o (PARI) {a(n) = if( n<1, 0, sigma(n))};
%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};
%o (PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* _Michael Somos_, Jan 29 2005 */
%o (PARI) max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ _Gottfried Helms_, Aug 10 2009
%o (MuPAD) numlib::sigma(n)$ n=1..81 // _Zerinvary Lajos_, May 13 2008
%o (SageMath) [sigma(n, 1) for n in range(1, 71)] # _Zerinvary Lajos_, Jun 04 2009
%o (Maxima) makelist(divsum(n),n,1,1000); \\ _Emanuele Munarini_, Mar 26 2011
%o (Haskell)
%o a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
%o -- _Reinhard Zumkeller_, May 07 2012
%o (Scheme) (definec (A000203 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (- (expt p (+ 1 e)) 1) (- p 1)) (A000203 (A028234 n)))))) ;; Uses macro definec from http://oeis.org/wiki/Memoization#Scheme - _Antti Karttunen_, Nov 25 2017
%o (Scheme) (define (A000203 n) (let ((r (sqrt n))) (let loop ((i (inexact->exact (floor r))) (s (if (integer? r) (- r) 0))) (cond ((zero? i) s) ((zero? (modulo n i)) (loop (- i 1) (+ s i (/ n i)))) (else (loop (- i 1) s)))))) ;; (Stand-alone program) - _Antti Karttunen_, Feb 20 2024
%o (GAP)
%o A000203:=List([1..10^2],n->Sigma(n)); # _Muniru A Asiru_, Oct 01 2017
%o (Python)
%o from sympy import divisor_sigma
%o def a(n): return divisor_sigma(n, 1)
%o print([a(n) for n in range(1, 71)]) # _Michael S. Branicky_, Jan 03 2021
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def a(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items())
%o print([a(n) for n in range(1, 51)]) # _Michael S. Branicky_, Feb 25 2024
%o (APL, Dyalog dialect) A000203 ← +/{ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð} ⍝ _Antti Karttunen_, Feb 20 2024
%Y See A034885, A002093 for records. Bisections give A008438, A062731. Values taken are listed in A007609. A054973 is an inverse function.
%Y For partial sums see A024916.
%Y Row sums of A127093.
%Y sigma_i (i=0..25): A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972, A281959.
%Y Cf. A144736, A158951, A158902, A174740, A147843, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A083728, A006352, A002659, A083238, A000593, A050449, A050452, A051731, A027748, A124010, A069192, A057641, A001318.
%Y Cf. A009194, A082062 (gcd(a(n),n) and its largest prime factor), A179931, A192795 (gcd(a(n),A001157(n)) and largest prime factor).
%Y Cf. also A034448 (sum of unitary divisors).
%Y Cf. A007955 (products of divisors).
%Y Cf. A144613, A144614, A144615, A146076.
%Y A001227, A000593 and this sequence have the same parity: A053866. - _Omar E. Pol_, May 14 2016
%Y Cf. A054533.
%K easy,core,nonn,nice,mult
%O 1,2
%A _N. J. A. Sloane_
|
