1,2

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).

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.

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).

Number of factors in the factorization of the polynomial x^n-1 over the integers. - T. D. Noe (noe(AT)sspectra.com), Apr 16 2003

If d(n) is odd, n is a perfect square. If d(n) = 2, n is prime. - Donald Sampson (Marsquo(AT)hotmail.com), Dec 10 2003

Number of even divisors of n = d(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2003

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 (g.resta(AT)iit.cnr.it), Feb 06 2006

Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2007

Sum_{n>0} d(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 15 2007

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 (rshepherd2(AT)hotmail.com), Apr 20 2008

Number of subgroups of the cyclic group of order n. - Benoit Jubin (benoit_jubin(AT)yahoo.fr), Apr 29 2008

Equals row sums of triangle A143319 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 26 2009: (Start)

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. (End)

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.

G. E. Andrews, Some debts I owe, Seminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. [From N. J. A. Sloane, Mar 12 2009]

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)

P. Erdos and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., 2 (1952), 257-271.

C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980.

K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)

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).

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.

E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.

E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.

Daniel Forgues, Table of n, a(n) for n=1..100000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Author?, Title?

G. E. Andrews, Some debts I owe

H. Bottomley, Illustration of initial terms

C. K. Caldwell, The Prime Glossary, Number of divisors

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

J. J. Holt & J. W. Jones, Discovering Number Theory, Section 1.4, Counting Divisors

M. Maia and M. Mendez, On the arithmetic product of combinatorial species

R. G. Martinez, Jr., The Factor Zone, Number of Factors for 1 through 600

Math Forum, Divisor Counting

K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n)

S. Ramanujan, On The Number Of Divisors Of A Number

H. B. Reiter, Counting Divisors

W. Sierpinski, Number Of Divisors And Their Sum

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function

Eric Weisstein's World of Mathematics, Binomial Number

Wikipedia, Table of divisors

Wolfram Research, Divisors of first 50 numbers

Index entries for "core" sequences

O. E. Pol, Illustration of initial terms (1) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009]

O. E. Pol, Illustration of initial terms (2) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009]

O. E. Pol, Illustration of initial terms (3) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009]

O. E. Pol, Illustration of initial terms (4) [From Omar E. Pol (info(AT)polprimos.com), Oct 25 2009]

O. E. Pol, Illustration of initial terms (5) [From Omar E. Pol (info(AT)polprimos.com), Oct 25 2009]

If n is written as 2^z*3^y*5^x*7^w*11^v*... then d(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...

Multiplicative with a(p^e) = e+1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f.: Sum_{n >= 1} d(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).

d(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].

a(n) is odd iff n is a square. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 29, 2001

a(n) = sum(k=1, n, f(k, n)) where f(k, n) = 1 if k divides n, 0 otherwise. 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 (ralf(AT)ark.in-berlin.de), Dec 25 2002

G.f.: Sum_{n>0} ((-1)^(n+1) x^(n(n+1)/2) / ((1-x^n)*Product(1-x^i, i=1..n))).

a(n)=n-sum(k=1, n, ceil(n/k)-floor(n/k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 11 2003

a(n) = A032741(n)+1 = A062011(n)/2 = A054519(n)-A054519(n-1) = A006218(n)-A006218(n-1) = sum(k=0, n-1, A051950(k)). - R. Stephan, Mar 26 2004

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

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 (qntmpkt(AT)yahoo.com), Apr 15 2007

A000005 = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divors of n, is a vector: [1, 1, 2, 1, 2, 2, 2,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007

Row sums of triangle A051731 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007

a(n)=sum(k=1, n, floor(n/k)-floor((n-1)/k) [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Aug 27 2009]

a(s)=2^omega(s), if s>1 is a squarefree number (A005117) and omega(s) is: A001221 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 08 2009]

with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];

a[n_] := DivisorSigma[0, n]

a[n_] := Length[Divisors[n]]

Table[Sum[Floor[n/k] - Floor[(n - 1)/k], {k, 1, n}], {n, 1, 100}] [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Aug 27 2009]

(PARI) a(n)=if(n<1, 0, numdiv(n))

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)^2)[n])

(MAGMA) [ NumberOfDivisors(n) : n in [1..100] ]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(MuPad)numlib::tau (n)$ n=1..90 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008

(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j); \\ log case

s=-log(prod(j=1, N, (1-x^j)^(1/j))); \\ A000005 the number of divisors of n.

s=serconvol(s, c)

v=Vec(s)

(Other) sage: [sigma(n, 0)for n in xrange(1, 105)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).

Cf. A001227, A005237, A005238, A006601, A006558, A019273, A039665, A049051.

Cf. A001826, A001842, A051731, A066446, A129510, A115361, A129372, A115361, A127093, A143319.

a(n) = A091220(A091202(n)). Cf. A061017.

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).

a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n). - Reinhard Zumkeller, May 08 2003

a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n). - Reinhard Zumkeller, May 08 2003

A159933, A159934 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 26 2009]

Cf. A027750, A163280. [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009]

Sequence in context: A074848 A167447 A134687 this_sequence A122667 A122668 A073668

Adjacent sequences: A000002 A000003 A000004 this_sequence A000006 A000007 A000008

easy,core,nonn,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).