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A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

%I A000005 M0246 N0086 #579 Aug 14 2025 21:55:02
%S A000005 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,
%T A000005 4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,
%U A000005 6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9,2,8,2,8
%N A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
%C A000005 If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k > 0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1) is the sum of the k-th powers of the divisors of n.
%C A000005 Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
%C A000005 Note that d(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
%C A000005 Number of factors in the factorization of the polynomial x^n-1 over the integers. - _T. D. Noe_, Apr 16 2003
%C A000005 Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e., max(p)=min(p). - _Giovanni Resta_, Feb 06 2006
%C A000005 Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - _Gary W. Adamson_, May 10 2007
%C A000005 For odd n, this is the number of partitions of n into consecutive integers. Proof: For n = 1, clearly true. For n = 2k + 1, k >= 1, map each (necessarily odd) divisor to such a partition as follows: For 1 and n, map k + (k+1) and n, respectively. For any remaining divisor d <= sqrt(n), map (n/d - (d-1)/2) + ... + (n/d - 1) + (n/d) + (n/d + 1) + ... + (n/d + (d-1)/2) {i.e., n/d plus (d-1)/2 pairs each summing to 2n/d}. For any remaining divisor d > sqrt(n), map ((d-1)/2 - (n/d - 1)) + ... + ((d-1)/2 - 1) + (d-1)/2 + (d+1)/2 + ((d+1)/2 + 1) + ... + ((d+1)/2 + (n/d - 1)) {i.e., n/d pairs each summing to d}. As all such partitions must be of one of the above forms, the 1-to-1 correspondence and proof is complete. - _Rick L. Shepherd_, Apr 20 2008
%C A000005 Number of subgroups of the cyclic group of order n. - _Benoit Jubin_, Apr 29 2008
%C A000005 Equals row sums of triangle A143319. - _Gary W. Adamson_, Aug 07 2008
%C A000005 Equals row sums of triangle A159934, equivalent to generating a(n) by convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2, ...) with the INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2, ...). Example: a(6) = 4 = (1, 1, 2, 2, 3, 2) dot (2, -1, 0, -1, 1, 1) = (2, -1, 0, -2, 3, 2) = 4. - _Gary W. Adamson_, Apr 26 2009
%C A000005 Number of times n appears in an n X n multiplication table. - _Dominick Cancilla_, Aug 02 2010
%C A000005 Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - _Juri-Stepan Gerasimov_, Oct 25 2015
%C A000005 The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - _Michael De Vlieger_, Dec 14 2016
%C A000005 a(n) is also the number of partitions of 2*n into equal parts, minus the number of partitions of 2*n into consecutive parts. - _Omar E. Pol_, May 03 2017
%C A000005 From _Tomohiro Yamada_, Oct 27 2020: (Start)
%C A000005 Let k(n) = log d(n)*log log n/(log 2 * log n), then lim sup k(n) = 1 (Hardy and Wright, Chapter 18, Theorem 317) and k(n) <= k(6983776800) = 1.537939... (the constant A280235) for every n (Nicolas and Robin, 1983).
%C A000005 There exist infinitely many n such that d(n) = d(n+1) (Heath-Brown, 1984). The number of such integers n <= x is at least c*x/(log log x)^3 (Hildebrand, 1987) but at most O(x/sqrt(log log x)) (Erdős, Carl Pomerance and Sárközy, 1987). (End)
%C A000005 Number of 2D grids of n congruent rectangles with two different side lengths, in a rectangle, modulo rotation (cf. A038548 for squares instead of rectangles). Also number of ways to arrange n identical objects in a rectangle (NOT modulo rotation, cf. A038548 for modulo rotation); cf. A007425 and A140773 for the 3D case. - _Manfred Boergens_, Jun 08 2021
%C A000005 The constant quoted above from Nicolas and Robin, 6983776800 = 2^5 * 3^3 * 5^2 * 7 * 11 * 13 * 17 * 19, appears arbitrary, but interestingly equals 2 * A095849(36). That second factor is highly composite and deeply composite. - _Hal M. Switkay_, Aug 08 2025
%D A000005 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 A000005 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A000005 G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
%D A000005 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 55.
%D A000005 G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.
%D A000005 K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
%D A000005 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
%D A000005 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. Has many references to this sequence. - _N. J. A. Sloane_, Jun 02 2014
%D A000005 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000005 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000005 B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1.
%D A000005 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 285.
%D A000005 E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
%D A000005 Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).
%H A000005 Daniel Forgues, <a href="/A000005/b000005.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from N. J. A. Sloane)
%H A000005 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, requires Flash plugin].
%H A000005 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
%H A000005 J. Bell, <a href="https://jordanbell.info/writing/2023/02/22/lambert-series-analytic-number-theory.html">Lambert series in analytic number theory</a>
%H A000005 R. Bellman and H. N. Shapiro, <a href="http://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., 49 (1948), 333-340. [From _N. J. A. Sloane_, Mar 12 2009]
%H A000005 Henry Bottomley, <a href="/A000005/a000005.gif">Illustration of initial terms</a>
%H A000005 D. M. Bressoud and M. V. Subbarao, <a href="http://dx.doi.org/10.4153/CMB-1984-022-5">On Uchimura's connection between partitions and the number of divisors</a>, Can. Math. Bull. 27, 143-145 (1984). Zbl 0536.10013.
%H A000005 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Tau">Number of divisors</a>
%H A000005 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 A000005 Jimmy Devillet and Gergely Kiss, <a href="https://arxiv.org/abs/1806.02073">Characterizations of biselective operations</a>, arXiv:1806.02073 [math.RA], 2018.
%H A000005 P. Erdős and L. Mirsky, <a href="http://www.renyi.hu/~p_erdos/1952-12.pdf">The distribution of values of the divisor function d(n)</a>, Proc. London Math. Soc. 2 (1952), pp. 257-271.
%H A000005 Paul Erdős, Carl Pomerance and András Sárközy, <a href="https://doi.org/10.1090/S0002-9939-1987-0897061-6">On locally repeated values of certain arithmetic functions, III</a>, Proc. Amer. Math. Soc. 101 (1987), 1-7.
%H A000005 C. R. Fletcher, <a href="http://www.jstor.org/stable/3615885">Rings of small order</a>, Math. Gaz. vol. 64, p. 13, 1980.
%H A000005 Robbert Fokkink and Jan van Neerven, <a href="https://www.nieuwarchief.nl/serie5/pdf/naw5-2003-04-3-269.pdf">Problemen/UWC</a> (in Dutch)
%H A000005 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 A000005 D. R. Heath-Brown, <a href="https://doi.org/10.1112/S0025579300010743">The divisor function at consecutive integers</a>, Mathematika 31 (1984), 141-149.
%H A000005 Adolf Hildebrand, <a href="https://projecteuclid.org/euclid.pjm/1102690578">The divisor function at consecutive integers</a>, Pacific J. Math. 129 (1987), 307-319.
%H A000005 J. J. Holt and J. W. Jones, <a href="https://web.archive.org/web/20190310134052/http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/divis4.html">Counting Divisors</a>, Discovering Number Theory, Section 1.4.
%H A000005 P. A. MacMahon, <a href="https://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 (1919), 75-113.
%H A000005 M. Maia and M. Mendez, <a href="https://arxiv.org/abs/math/0503436">On the arithmetic product of combinatorial species</a>, arXiv:math/0503436 [math.CO], 2005.
%H A000005 R. G. Martinez, Jr., The Factor Zone, <a href="http://factorzone.tripod.com/factors.htm">Number of Factors for 1 through 600</a>.
%H A000005 Math Forum, <a href="http://mathforum.org/library/drmath/view/55741.html">Divisor Counting</a>.
%H A000005 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3213216/a-question-on-discrete-fourier-transform-of-some-function">A question on discrete Fourier Transform of some function</a>
%H A000005 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 A000005 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.009">A new look on the generating function for the number of divisors</a>, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
%H A000005 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 2.1.
%H A000005 J. L. Nicolas and G. Robin, <a href="https://doi.org/10.4153/CMB-1983-078-5">Majorations Explicites Pour le Nombre de Diviseurs de N</a>, Canadian Mathematical Bulletin, Volume 26, Issue 4, 01 December 1983, pp. 485-492.
%H A000005 Matthew Parker, <a href="https://oeis.org/A000005/a000005_25M.7z">The first 25 million terms (7-Zip compressed file)</a>.
%H A000005 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H A000005 Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
%H A000005 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Illustration of initial terms: figure 1</a>, <a href="http://www.polprimos.com/imagenespub/poldiv02.jpg">figure 2</a>, <a href="http://www.polprimos.com/imagenespub/poldiv03.jpg">figure 3</a>, <a href="http://www.polprimos.com/imagenespub/poldiv04.jpg">figure 4</a>, <a href="http://www.polprimos.com/imagenespub/poldiv3v.jpg">figure 5</a>, (2009), <a href="http://www.polprimos.com/imagenespub/poldiv13.jpg">figure 6 (a, b, c)</a>, (2013)
%H A000005 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper8/page1.htm">On The Number Of Divisors Of A Number</a>.
%H A000005 H. B. Reiter, <a href="https://webpages.charlotte.edu/~hbreiter/m6105/Divisors.pdf">Counting Divisors</a>.
%H A000005 W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>.
%H A000005 Terence Tao, <a href="https://terrytao.wordpress.com/wp-content/uploads/2009/01/whatsnew.pdf">Poincaré's Legacies: pages from year two of a mathematical blog</a>, see page 59.
%H A000005 E. C. Titchmarsh, <a href="https://doi.org/10.1112/jlms/s1-13.4.248">On a series of Lambert type</a>, J. London Math. Soc., 13 (1938), 248-253.
%H A000005 Keisuke Uchimura, <a href="http://dx.doi.org/10.1016/0097-3165(81)90009-1">An identity for the divisor generating function arising from sorting theory</a>, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
%H A000005 Wang Zheng Bing, Robert Fokkink and Wan Fokkink, <a href="http://www.jstor.org/stable/2974956">A Relation Between Partitions and the Number of Divisors</a>, Am. Math. Monthly, 102 (Apr., 1995), no. 4, 345-347.
%H A000005 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialNumber.html">Binomial Number</a>, <a href="https://mathworld.wolfram.com/DirichletSeriesGeneratingFunction.html">Dirichlet Series Generating Function</a>, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>, and <a href="https://mathworld.wolfram.com/MoebiusTransform.html">Moebius Transform</a>.
%H A000005 Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.
%H A000005 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02">Divisors of first 50 numbers</a>
%H A000005 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000005 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A000005 If n is written as 2^z*3^y*5^x*7^w*11^v*... then a(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
%F A000005 a(n) = 2 iff n is prime.
%F A000005 G.f.: Sum_{n >= 1} a(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
%F A000005 a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
%F A000005 a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
%F A000005 Multiplicative with a(p^e) = e+1. - _David W. Wilson_, Aug 01 2001
%F A000005 a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
%F A000005 a(n) is odd iff n is a square. - _Reinhard Zumkeller_, Dec 29 2001
%F A000005 a(n) = Sum_{k=1..n} f(k, n) where f(k, n) = 1 if k divides n, 0 otherwise (Mobius transform of A000012). Equivalently, f(k, n) = (1/k)*Sum_{l=1..k} z(k, l)^n with z(k, l) the k-th roots of unity. - _Ralf Stephan_, Dec 25 2002
%F A000005 G.f.: Sum_{k>0} ((-1)^(k+1) * x^(k * (k + 1)/2) / ((1 - x^k) * Product_{i=1..k} (1 - x^i))). - _Michael Somos_, Apr 27 2003
%F A000005 a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - _Benoit Cloitre_, May 11 2003
%F A000005 a(n) = A032741(n) + 1 = A062011(n)/2 = A054519(n) - A054519(n-1) = A006218(n) - A006218(n-1) = 1 + Sum_{k=1..n-1} A051950(k+1). - _Ralf Stephan_, Mar 26 2004
%F A000005 G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - _Michael Somos_, Apr 05 2003
%F A000005 Sequence = M*V where M = A129372 as an infinite lower triangular matrix and V = ruler sequence A001511 as a vector: [1, 2, 1, 3, 1, 2, 1, 4, ...]. - _Gary W. Adamson_, Apr 15 2007
%F A000005 Sequence = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divisors of n, is a vector: [1, 1, 2, 1, 2, 2, 2, ...]. - _Gary W. Adamson_, Apr 15 2007
%F A000005 Row sums of triangle A051731. - _Gary W. Adamson_, Nov 02 2007
%F A000005 Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - _Gerald McGarvey_, Dec 15 2007
%F A000005 Logarithmic g.f.: Sum_{n>=1} a(n)/n * x^n = -log( Product_{n>=1} (1-x^n)^(1/n) ). - _Joerg Arndt_, May 03 2008
%F A000005 a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - _Enrique Pérez Herrero_, Aug 27 2009
%F A000005 a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - _Enrique Pérez Herrero_, Sep 08 2009
%F A000005 a(n) = A048691(n) - A055205(n). - _Reinhard Zumkeller_, Dec 08 2009
%F A000005 For n > 1, a(n) = 2 + Sum_{k=2..n-1} floor((cos(Pi*n/k))^2). And floor((cos(Pi*n/k))^2) = floor(1/4 * e^(-(2*i*Pi*n)/k) + 1/4 * e^((2*i*Pi*n)/k) + 1/2). - _Eric Desbiaux_, Mar 09 2010, corrected Apr 16 2011
%F A000005 a(n) = 1 + Sum_{k=1..n} (floor(2^n/(2^k-1)) mod 2) for every n. - Fabio Civolani (civox(AT)tiscali.it), Mar 12 2010
%F A000005 From _Vladimir Shevelev_, May 22 2010: (Start)
%F A000005 (Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
%F A000005 Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
%F A000005 a(n) = sigma_0(n) = 1 + Sum_{m>=2} Sum_{r>=1} (1/m^(r+1))*Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} e^(2*k*Pi*i*(n+(m-j)*m^r)/m^(r+1)). - _A. Neves_, Oct 04 2010
%F A000005 a(n) = 2*A038548(n) - A010052(n). - _Reinhard Zumkeller_, Mar 08 2013
%F A000005 Sum_{n>=1} a(n)*q^n = (log(1-q) + psi_q(1)) / log(q), where psi_q(z) is the q-digamma function. - _Vladimir Reshetnikov_, Apr 23 2013
%F A000005 a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - _Reinhard Zumkeller_, Jul 12 2013
%F A000005 a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - _Mircea Merca_, Feb 18 2013
%F A000005 G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - _Mats Granvik_, Jun 15 2013
%F A000005 The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - _Joerg Arndt_, Mar 12 2014
%F A000005 G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - _Paul D. Hanna_, Aug 21 2014
%F A000005 2*Pi*a(n) = Sum_{m=1..n} Integral_{x=0..2*Pi} r^(m-n)( cos((m-n)*x)-r^m cos(n*x) )/( 1+r^(2*m)-2r^m cos(m*x) )dx, 0 < r < 1 a free parameter. This formula is obtained as the sum of the residues of the Lambert series Sum_{k>=1} x^k/(1-x^k). - _Seiichi Kirikami_, Oct 22 2015
%F A000005 a(n) = A091220(A091202(n)) = A106737(A156552(n)). - _Antti Karttunen_, circa 2004 & Mar 06 2017
%F A000005 a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - _George Beck_, May 06 2017
%F A000005 G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - _Mamuka Jibladze_, Aug 29 2018
%F A000005 a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - _Daniel Suteu_, Nov 05 2018
%F A000005 a(k*n) = a(n)*(f(k,n)+2)/(f(k,n)+1), where f(k,n) is the exponent of the highest power of k dividing n and k is prime. - _Gary Detlefs_, Feb 08 2019
%F A000005 a(n) = 2*log(p(n))/log(n), n > 1, where p(n)= the product of the factors of n = A007955(n). - _Gary Detlefs_, Feb 15 2019
%F A000005 a(n) = (1/n) * Sum_{k=1..n} sigma(gcd(n,k)), where sigma(n) = sum of divisors of n. - _Orges Leka_, May 09 2019
%F A000005 a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - _Ivan N. Ianakiev_, Nov 14 2019
%F A000005 From _Richard L. Ollerton_, May 11 2021: (Start)
%F A000005 a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
%F A000005 a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
%F A000005 From _Ridouane Oudra_, Nov 12 2021: (Start)
%F A000005 a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
%F A000005 a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)
%e A000005 G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + ...
%p A000005 with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];
%t A000005 Table[DivisorSigma[0, n], {n, 100}] (* _Enrique Pérez Herrero_, Aug 27 2009 *)
%t A000005 CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* _Vladimir Reshetnikov_, Apr 23 2013 *)
%t A000005 a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* _Michael Somos_, Apr 25 2013 *)
%t A000005 a[ n_] := SeriesCoefficient[ q/(1 - q)^2 QHypergeometricPFQ[ {q, q}, {q^2, q^2}, q, q^2], {q, 0, Abs@n}]; (* _Michael Somos_, Mar 05 2014 *)
%t A000005 a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* _Mats Granvik_, Apr 15 2015 *)
%t A000005 With[{M=500},CoefficientList[Series[(2x)/(1-x)-Sum[x^k (1-2x^k)/(1-x^k),{k,M}],{x,0,M}],x]] (* _Mamuka Jibladze_, Aug 31 2018 *)
%o A000005 (PARI) {a(n) = if( n==0, 0, numdiv(n))}; /* _Michael Somos_, Apr 27 2003 */
%o A000005 (PARI) {a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* _Michael Somos_, Apr 27 2003 */
%o A000005 (PARI) {a(n)=polcoeff(sum(m=1, n+1, sumdiv(m, d, (-log(1-x^(m/d) +x*O(x^n) ))^d/d!)), n)} \\ _Paul D. Hanna_, Aug 21 2014
%o A000005 (Magma) [ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o A000005 (MuPAD) numlib::tau (n)$ n=1..90 // _Zerinvary Lajos_, May 13 2008
%o A000005 (Sage) [sigma(n, 0) for n in range(1, 105)]  # _Zerinvary Lajos_, Jun 04 2009
%o A000005 (Haskell)
%o A000005 divisors 1 = [1]
%o A000005 divisors n = (1:filter ((==0) . rem n)
%o A000005                [2..n `div` 2]) ++ [n]
%o A000005 a = length . divisors
%o A000005 -- _James Spahlinger_, Oct 07 2012
%o A000005 (Haskell)
%o A000005 a000005 = product . map (+ 1) . a124010_row  -- _Reinhard Zumkeller_, Jul 12 2013
%o A000005 (Python)
%o A000005 from sympy import divisor_count
%o A000005 for n in range(1, 20): print(divisor_count(n), end=', ') # _Stefano Spezia_, Nov 05 2018
%o A000005 (GAP) List([1..150],n->Tau(n)); # _Muniru A Asiru_, Mar 05 2019
%o A000005 (Julia)
%o A000005 function tau(n)
%o A000005     i = 2; num = 1
%o A000005     while i * i <= n
%o A000005         if rem(n, i) == 0
%o A000005             e = 0
%o A000005             while rem(n, i) == 0
%o A000005                 e += 1
%o A000005                 n = div(n, i)
%o A000005             end
%o A000005             num *= e + 1
%o A000005         end
%o A000005         i += 1
%o A000005     end
%o A000005     return n > 1 ? num + num : num
%o A000005 end
%o A000005 println([tau(n) for n in 1:104])  # _Peter Luschny_, Sep 03 2023
%Y A000005 See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
%Y A000005 For partial sums see A006218.
%Y A000005 Cf. A007427 (Dirichlet Inverse), A001227, A001826, A001842, A005237, A005238, A006558, A006601, A019273, A027750, A034296, A039665, A049051, A049820, A051731, A061017, A066446, A091202, A091220, A106737, A115361, A127093, A129372, A129510, A143319, A156552, A159933, A159934, A163280, A183063, A237665, A263730.
%Y A000005 Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
%Y A000005 Cf. A000010, A095849.
%Y A000005 Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).
%K A000005 nonn,easy,core,nice,mult,hear
%O A000005 1,2
%A A000005 _N. J. A. Sloane_
%E A000005 Incorrect formula deleted by _Ridouane Oudra_, Oct 28 2021