Search: id:A000005 Results 1-1 of 1 results found. %I A000005 M0246 N0086 %S A000005 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4, 9, %T A000005 2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8, 2, %U A000005 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). %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 (noe(AT)sspectra.com), Apr 16 2003 %C A000005 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 %C A000005 Number of even divisors of n = d(2*n) * (1 - n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 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 (g.resta(AT)iit.cnr.it), 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 (qntmpkt(AT)yahoo.com), May 10 2007 %C A000005 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 %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 (rshepherd2(AT)hotmail.com), Apr 20 2008 %C A000005 Number of subgroups of the cyclic group of order n. - Benoit Jubin (benoit_jubin(AT)yahoo.fr), Apr 29 2008 %C A000005 Equals row sums of triangle A143319 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2008] %C A000005 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 26 2009: (Start) %C A000005 Equals row sums of triangle A159934, equivalent to generating a(n) by %C A000005 convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2,...) with the %C A000005 INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2,...): %C A000005 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) %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 G. E. Andrews, Some debts I owe, Seminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000; see (7.1). %D A000005 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38. %D A000005 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] %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 P. Erdos and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., 2 (1952), 257-271. %D A000005 C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980. %D A000005 K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451. %D A000005 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113. %D A000005 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.) %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 E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160. %D A000005 E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253. %H A000005 Daniel Forgues, Table of n, a(n) for n=1..100000 %H A000005 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]. %H A000005 Author?, Title? %H A000005 G. E. Andrews, Some debts I owe %H A000005 H. Bottomley, Illustration of initial terms %H A000005 C. K. Caldwell, The Prime Glossary, Number of divisors %H A000005 Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003. %H A000005 J. J. Holt & J. W. Jones, Discovering Number Theory, Section 1.4, Counting Divisors %H A000005 M. Maia and M. Mendez, On the arithmetic product of combinatorial species %H A000005 R. G. Martinez, Jr., The Factor Zone, Number of Factors for 1 through 600 %H A000005 Math Forum, Divisor Counting %H A000005 K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n) %H A000005 S. Ramanujan, On The Number Of Divisors Of A Number %H A000005 H. B. Reiter, Counting Divisors %H A000005 W. Sierpinski, Number Of Divisors And Their Sum %H A000005 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1). %H A000005 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2). %H A000005 Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function %H A000005 Eric Weisstein's World of Mathematics, Binomial Number %H A000005 Wikipedia, Table of divisors %H A000005 Wolfram Research, Divisors of first 50 numbers %H A000005 Index entries for "core" sequences %H A000005 O. E. Pol, Illustration of initial terms (1) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009] %H A000005 O. E. Pol, Illustration of initial terms (2) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009] %H A000005 O. E. Pol, Illustration of initial terms (3) [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009] %H A000005 O. E. Pol, Illustration of initial terms (4) [From Omar E. Pol (info(AT)polprimos.com), Oct 25 2009] %H A000005 O. E. Pol, Illustration of initial terms (5) [From Omar E. Pol (info(AT)polprimos.com), Oct 25 2009] %F A000005 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)*... %F A000005 Multiplicative with a(p^e) = e+1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001. %F A000005 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). %F A000005 d(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522]. %F A000005 a(n) is odd iff n is a square. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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. 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 %F A000005 G.f.: Sum_{n>0} ((-1)^(n+1) x^(n(n+1)/2) / ((1-x^n)*Product(1-x^i, i=1..n))). %F A000005 a(n)=n-sum(k=1, n, ceil(n/k)-floor(n/k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 11 2003 %F A000005 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 %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 (qntmpkt(AT)yahoo.com), Apr 15 2007 %F A000005 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 %F A000005 Row sums of triangle A051731 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007 %F A000005 a(n)=sum(k=1, n, floor(n/k)-floor((n-1)/k) [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Aug 27 2009] %F A000005 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] %p A000005 with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ]; %t A000005 a[n_] := DivisorSigma[0, n] %t A000005 a[n_] := Length[Divisors[n]] %t A000005 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] %o A000005 (PARI) a(n)=if(n<1,0,numdiv(n)) %o A000005 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2)[n]) %o A000005 (MAGMA) [ NumberOfDivisors(n) : n in [1..100] ]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006 %o A000005 (MuPad)numlib::tau (n)$ n=1..90 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008 %o A000005 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008) %o A000005 N=17; default(seriesprecision,N); x=z+O(z^(N+1)) %o A000005 c=sum(j=1,N,j*x^j); \\ log case %o A000005 s=-log(prod(j=1,N,(1-x^j)^(1/j))); \\ A000005 the number of divisors of n. %o A000005 s=serconvol(s,c) %o A000005 v=Vec(s) %o A000005 (Other) sage: [sigma(n,0)for n in xrange(1,105)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009] %Y A000005 See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n). %Y A000005 Cf. A001227, A005237, A005238, A006601, A006558, A019273, A039665, A049051. %Y A000005 Cf. A001826, A001842, A051731, A066446, A129510, A115361, A129372, A115361, A127093, A143319. %Y A000005 a(n) = A091220(A091202(n)). Cf. A061017. %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 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 %Y A000005 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 %Y A000005 A159933, A159934 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 26 2009] %Y A000005 Cf. A027750, A163280. [From Omar E. Pol (info(AT)polprimos.com), Oct 22 2009] %Y A000005 Sequence in context: A074848 A167447 A134687 this_sequence A122667 A122668 A073668 %Y A000005 Adjacent sequences: A000002 A000003 A000004 this_sequence A000006 A000007 A000008 %K A000005 easy,core,nonn,nice,mult %O A000005 1,2 %A A000005 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.008 seconds