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

Number of even divisors of n = a(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.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

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.

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)

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.

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.

N. J. A. Sloane, Table of n, d(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 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

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).

a(n) is odd iff n is a square. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.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

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

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

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

(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

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

See A002183, A002182 for records.

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

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

Cf. A001826, A001842.

Cf. A129372, A115361.

Cf. A066446, A129510, A001227, A115361.

Cf. A127093.

Cf. A051731.

Adjacent sequences: A000002 A000003 A000004 this_sequence A000006 A000007 A000008

Sequence in context: A035149 A074848 A134687 this_sequence A122667 A122668 A073668

easy,core,nonn,nice,mult,new

njas