A002183 -id:A002183 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

N. J. A. Sloane

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.
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, Apr 16 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, Feb 06 2006
Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 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, Apr 20 2008
Number of subgroups of the cyclic group of order n. - Benoit Jubin, Apr 29 2008
Equals row sums of triangle A143319. - Gary W. Adamson, Aug 07 2008
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
Number of times n appears in an n X n multiplication table. - Dominick Cancilla, Aug 02 2010
Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - Juri-Stepan Gerasimov, Oct 25 2015
The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - Michael De Vlieger, Dec 14 2016
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
From Tomohiro Yamada, Oct 27 2020: (Start)
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).
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)
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
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

			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 + ...

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.
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)
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 55.
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.
K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
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
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.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 285.
E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
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, requires Flash plugin].
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
J. Bell, Lambert series in analytic number theory
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]
Henry Bottomley, Illustration of initial terms
D. M. Bressoud and M. V. Subbarao, On Uchimura's connection between partitions and the number of divisors, Can. Math. Bull. 27, 143-145 (1984). Zbl 0536.10013.
C. K. Caldwell, The Prime Glossary, Number of divisors
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
Jimmy Devillet and Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.
P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), pp. 257-271.
Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions, III, Proc. Amer. Math. Soc. 101 (1987), 1-7.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980.
Robbert Fokkink and Jan van Neerven, Problemen/UWC (in Dutch)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129 (1987), 307-319.
J. J. Holt and J. W. Jones, Counting Divisors, Discovering Number Theory, Section 1.4.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
R. G. Martinez, Jr., The Factor Zone, Number of Factors for 1 through 600.
Math Forum, Divisor Counting.
Mathematics Stack Exchange, A question on discrete Fourier Transform of some function
K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 2.1.
J. L. Nicolas and G. Robin, Majorations Explicites Pour le Nombre de Diviseurs de N, Canadian Mathematical Bulletin, Volume 26, Issue 4, 01 December 1983, pp. 485-492.
Matthew Parker, The first 25 million terms (7-Zip compressed file).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
Omar E. Pol, Illustration of initial terms: figure 1, figure 2, figure 3, figure 4, figure 5, (2009), figure 6 (a, b, c), (2013)
S. Ramanujan, On The Number Of Divisors Of A Number.
H. B. Reiter, Counting Divisors.
W. Sierpiński, Number Of Divisors And Their Sum.
Terence Tao, Poincaré's Legacies: pages from year two of a mathematical blog, see page 59.
E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.
Keisuke Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
Wang Zheng Bing, Robert Fokkink and Wan Fokkink, A Relation Between Partitions and the Number of Divisors, Am. Math. Monthly, 102 (Apr., 1995), no. 4, 345-347.
Eric Weisstein's World of Mathematics, Binomial Number, Dirichlet Series Generating Function, Divisor Function, and Moebius Transform.
Wikipedia, Table of divisors.
Wolfram Research, Divisors of first 50 numbers
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
For partial sums see A006218.
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.
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).
Cf. A000010, A095849.
Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).

List([1..150],n->Tau(n)); # Muniru A Asiru, Mar 05 2019

divisors 1 = [1]
divisors n = (1:filter ((==0) . rem n)
               [2..n `div` 2]) ++ [n]
a = length . divisors
-- James Spahlinger, Oct 07 2012

a000005 = product . map (+ 1) . a124010_row  -- Reinhard Zumkeller, Jul 12 2013

function tau(n)
    i = 2; num = 1
    while i * i <= n
        if rem(n, i) == 0
            e = 0
            while rem(n, i) == 0
                e += 1
                n = div(n, i)
            end
            num *= e + 1
        end
        i += 1
    end
    return n > 1 ? num + num : num
end
println([tau(n) for n in 1:104])  # Peter Luschny, Sep 03 2023

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

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

Table[DivisorSigma[0, n], {n, 100}] (* Enrique Pérez Herrero, Aug 27 2009 *)
CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* Vladimir Reshetnikov, Apr 23 2013 *)
a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* Michael Somos, Apr 25 2013 *)
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 *)
a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* Mats Granvik, Apr 15 2015 *)
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 *)

numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, numdiv(n))}; /* Michael Somos, Apr 27 2003 */

{a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* Michael Somos, Apr 27 2003 */

{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

from sympy import divisor_count
for n in range(1, 20): print(divisor_count(n), end=', ') # Stefano Spezia, Nov 05 2018

[sigma(n, 0) for n in range(1, 105)]  # Zerinvary Lajos, Jun 04 2009

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)*...
a(n) = 2 iff n is prime.
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).
a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
Multiplicative with a(p^e) = e+1. - David W. Wilson, Aug 01 2001
a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
a(n) is odd iff n is a square. - Reinhard Zumkeller, Dec 29 2001
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
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
a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - Benoit Cloitre, May 11 2003
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
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, Apr 15 2007
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
Row sums of triangle A051731. - Gary W. Adamson, Nov 02 2007
Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - Gerald McGarvey, Dec 15 2007
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
a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - Enrique Pérez Herrero, Aug 27 2009
a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - Enrique Pérez Herrero, Sep 08 2009
a(n) = A048691(n) - A055205(n). - Reinhard Zumkeller, Dec 08 2009
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
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
From Vladimir Shevelev, May 22 2010: (Start)
(Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
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
a(n) = 2*A038548(n) - A010052(n). - Reinhard Zumkeller, Mar 08 2013
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
a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - Reinhard Zumkeller, Jul 12 2013
a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - Mircea Merca, Feb 18 2013
G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - Mats Granvik, Jun 15 2013
The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - Joerg Arndt, Mar 12 2014
G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - Paul D. Hanna, Aug 21 2014
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
a(n) = A091220(A091202(n)) = A106737(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - George Beck, May 06 2017
G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - Mamuka Jibladze, Aug 29 2018
a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - Daniel Suteu, Nov 05 2018
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
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
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
a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - Ivan N. Ianakiev, Nov 14 2019
From Richard L. Ollerton, May 11 2021: (Start)
a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
From Ridouane Oudra, Nov 12 2021: (Start)
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)

Incorrect formula deleted by Ridouane Oudra, Oct 28 2021

A002182 Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160

Offset: 1

N. J. A. Sloane

Where record values of d(n) occur: d(n) > d(k) for all k < n.
A002183 is the RECORDS transform of A000005, i.e., lists the corresponding values d(n) for n in A002182.
Flammenkamp's page also has a copy of the missing Siano paper.
Highly composite numbers are the product of primorials, A002110. See A112779 for the number of primorial terms in the product of a highly composite number. - Jud McCranie, Jun 12 2005
Sigma and tau for highly composite numbers through the 146th entry conform to a power fit as follows: log(sigma)=A*log(tau)^B where (A,B) =~ (1.45,1.38). - Bill McEachen, May 24 2006
a(n) often corresponds to P(n,m) = number of permutations of n things taken m at a time. Specifically, if start=1, pointers 1-6, 9, 10, 13-15, 17-19, 22, 23, 28, 34, 37, 43, 52, ... An example is a(37)=665280, which is P(12,6)=12!/(12-6)!. - Bill McEachen, Feb 09 2009
Concerning the previous comment, if m=1, then P(n,m) can represent any number. So let's assume m > 1. Searching the first 1000 terms, the only indices of terms of the form P(n,m) are 4, 5, 6, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 27, 28, 31, 34, 37, 41, 43, 44, 47, 50, 52, and 54. Note that a(44) = 4324320 = P(2079,2). See A163264. - T. D. Noe, Jun 10 2009
A large number of highly composite numbers have 9 as their digit root. - Parthasarathy Nambi, Jun 07 2009
Because 9 divides all highly composite numbers greater than 1680, those numbers have digital root 9. - T. D. Noe, Jul 24 2009
See A181309 for highly composite numbers that are not highly abundant.
a(n) is also defined by the recurrence: a(1) = 1, a(n+1)/sigma(a(n+1)) < a(n) / sigma(a(n)). - Michel Lagneau, Jan 02 2012 [NOTE: This "definition" is wrong (a(20)=7560 does not satisfy this inequality) and incomplete: It does not determine a sequence uniquely, e.g., any subsequence would satisfy the same relation. The intended meaning is probably the definition of the (different) sequence A004394. - M. F. Hasler, Sep 13 2012]
Up to a(1000), the terms beyond a(5) = 12 resp. beyond a(9) = 60 are a multiples of these. Is this true for all subsequent terms? - M. F. Hasler, Sep 13 2012 [Yes: see EXAMPLE in A199337! - M. F. Hasler, Jan 03 2020]
Differs from the superabundant numbers from a(20)=7560 on, which is not in A004394. The latter is not a subsequence of A002182, as might appear from considering the displayed terms: The two sequences have only 449 terms in common, the largest of which is A002182(2567) = A004394(1023). See A166735 for superabundant numbers that are not highly composite, and A004394 for further information. - M. F. Hasler, Sep 13 2012
Subset of A067128 and of A025487. - David A. Corneth, May 16 2016, Jan 03 2020
It seems that a(n) +- 1 is often prime. For n <= 1000 there are 210 individual primes and 17 pairs of twin primes. See link to Lim's paper below. - Dmitry Kamenetsky, Mar 02 2019
There are infinitely many numbers in this sequence and a(n+1) <= 2*a(n), because it is sufficient to multiply a(n) by 2 to get a number having more divisors. (This proves Guess 0 in the Lim paper.) For n = (1, 2, 4, 5, 9, 13, 18, ...) one has equality in this bound, but asymptotically a(n+1)/a(n) goes to 1, cf. formula due to Erdős. See A068507 for the terms such that a(n)+-1 are twin primes. - M. F. Hasler, Jun 23 2019
Conjecture: For n > 7, a(n) is a Zumkeller number (A083207). Verified for n up to and including 48. If this conjecture is true, one may base on it an alternative proof of the fact that for n>7 a(n) is not a perfect square (see Fact 5, Rao/Peng arXiv link at A083207). - Ivan N. Ianakiev, Jun 29 2019
The conjecture above is true (see the proof in the "Links" section). - Ivan N. Ianakiev, Jan 31 2020
The first instance of omega(a(n)) < omega(a(n-1)) (omega = A001221: number of prime divisors) is at a(26) = 45360. Up to n = 10^4, 1759 terms have this property, but omega decreases by 2 only at indices n = 5857, 5914 and 5971. - M. F. Hasler, Jan 02 2020
Inequality (54) in Ramanujan (1915) implies that for any m there is n* such that m | a(n) for all n > n*: see A199337 for the proof. - M. F. Hasler, Jan 03 2020

			a(5) = 12 is in the sequence because A000005(12) is larger than any earlier value in A000005. - _M. F. Hasler_, Jan 03 2020

CRC Press Standard Mathematical Tables, 28th Ed, p. 61.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris 2008.
L. E. Dickson, History of Theory of Numbers, I, p. 323.
Ross Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985
Jean-Louis Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 88.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (obtained from A. Flammenkamp's data; first 1000 terms from T. D. Noe)
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Harold W. Ellingsen, Jr., A Fresh Look at Highly Composite Numbers, The American Mathematical Monthly, Vol. 126, No. 8 (2019), pp. 740-741.
Paul Erdős, On Highly composite numbers, J. London Math. Soc., Vol. 19 (1944), pp. 130-133, MR7,145d; Zentralblatt 61,79.
Achim Flammenkamp, Highly composite numbers.
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Achim Flammenkamp, List of the first 779,674 highly composite numbers.
James Grime and Brady Haran, 5040 and other Anti-Prime Numbers, Numberphile video (2016).
Bob Hinman, Letter to N. J. A. Sloane, Aug. 1980.
Ivan N. Ianakiev, On the question "Which Highly composite numbers (A002182) are Zumkeller numbers (A083207)?".
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
Aneesh M. Koya and P. P. Deepthi, Plug and play self-configurable IoT gateway node for telemonitoring of ECG, Computers in Biology and Medicine, Vol. 112 (2019), 103359.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543; arXiv:math/0008177 [math.NT], 2000-2001.
Bill Lauritzen, Versatile Numbers -Versatile Economics.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola Magazine, Volume 54, Issue 3, (2018).
R. J. Mathar, Maple program to convert the Flammenkamp file to an OEIS b-file.
R. J. Mathar, Output of above Maple program. [Uncompresses to 9.1 MB]
Graeme McRae, Highly Composite Numbers.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers' (Text in French).
Jean-Louis Nicolas and Guy Robin, Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
Kevin O'Bryant, PlanetMath.org, Highly composite number.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915), pp. 347-409. (DOI: 10.1112/plms/s2_14.1.347, also available with an additional footnote in the PDF at http://ramanujan.sirinudi.org/Volumes/published/ram15.html)
Steven Ratering, An interesting subset of the highly composite numbers, Math. Mag., Vol. 64, No. 5 (1991), pp. 343-346.
Guy Robin, Méthodes d'optimisation pour un problème de théorie des nombres, RAIRO Informatique Théorique, Vol. 17, No. 3 (1983), pp. 239-247.
Vladimir Shevelev, On Erdős constant, arXiv:1605.08884 [math.NT], 2016.
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.
N. J. A. Sloane, Transforms.
Michel Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Highly Composite Number.
Wikipedia, Highly composite number.

Cf. A000005 (number of divisors), A002110, A002183, A002473, A004394, A025487, A106037, A108602, A112778, A112779, A112780, A112781, A006218, A126098, A002201, A072938, A094348, A003418, A161184, A037992 (infinitary analog), A108951, A329902, A352418.
Cf. A261100 (a left inverse).
Cf. A002808. - Peter J. Marko, Aug 16 2018
Cf. A279930 (highly composite and highly Brazilian).
Cf. A068507 (terms such that a(n)+-1 are twin primes).
Cf. A199337 (number of terms not divisible by n).

a = 0; Do[b = DivisorSigma[0, n]; If[b > a, a = b; Print[n]], {n, 1, 10^7}]
(* Convert A. Flammenkamp's 779674-term dataset; first, decompress, rename "HCN.txt": *)
a = Times @@ {Times @@ Prime@ Range@ ToExpression@ First@ #1, If[# == {}, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, #]] &@ DeleteCases[-1 + Flatten@ Map[If[StringFreeQ[#, "^"], ToExpression@ #, ConstantArray[#1, #2] & @@ ToExpression@ StringSplit[#, "^"]] &, #2], 0]} & @@ TakeDrop[StringSplit@ #, 1] & /@ Import["HCN.txt", "Data"] (* Michael De Vlieger, May 08 2018 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,2163000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[All,1]] (* Harvey P. Dale, May 13 2022 *)
NestList[Function[last,
  Module[{d = DivisorSigma[0, last]},
   NestWhile[# + 1 &, last, DivisorSigma[0, #] <= d &]]], 1, 40] (* Steven Lu, Mar 30 2023 *)

print1(r=1); forstep(n=2,1e5,2, if(numdiv(n)>r, r=numdiv(n); print1(", "n))) \\ Charles R Greathouse IV, Jun 10 2011

v002182 = [1]/*vector for memoization*/; A002182(n, i = #v002182) ={ if(n > i, v002182 = Vec(v002182, n); my(k = v002182[i], d, s=1); until(i == n, d = numdiv(k); s<60 && k>=60 && s=60; until(numdiv(k += s) > d,); v002182[i++] = k); k, v002182[n])} \\ Antti Karttunen, Jun 06 2017; edited by M. F. Hasler, Jan 03 2020 and Jun 20 2022

is_A002182(n, a=1, b=1)={while(n>A002182(b*=2), a*=2); until(a>b, my(m=(a+b)\2, t=A002182(m)); if(tn, b=m-1, return(m)))} \\ Also used in other sequences. - M. F. Hasler, Jun 20 2022

from sympy import divisor_count
A002182_list, r = [], 0
for i in range(1,10**4):
    d = divisor_count(i)
    if d > r:
        r = d
        A002182_list.append(i) # Chai Wah Wu, Mar 23 2015

Also, for n >= 2, smallest values of p for which A006218(p)-A006318(p-1) = A002183(n). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n+1) < a(n) * (1+log(a(n))^-c) for some positive c (see Erdős). - David A. Corneth, May 16 2016
a(n) = A108951(A329902(n)). - Antti Karttunen, Jan 08 2020
a(n+1) <= 2*a(n). For cases where the equal sign holds, see A072938. - A.H.M. Smeets, Jul 10 2021
Sum_{n>=1} 1/a(n) = A352418. - Amiram Eldar, Mar 24 2022

Jun 19 1996: Changed beginning to start at 1.
Jul 10 1996: Matthew Conroy points out that these are different from the super-abundant numbers - see A004394. Last 8 terms sent by J. Lowell; checked by Jud McCranie.
Description corrected by Gerard Schildberger and N. J. A. Sloane, Apr 04 2001
Additional references from Lekraj Beedassy, Jul 24 2001

A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405

Offset: 1

Eric W. Weisstein

Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i*3^j*5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra.
Numbers k such that 8*k = EulerPhi(30*k). - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A165704: A165705(n) = A165704(a(n)). - Reinhard Zumkeller, Sep 26 2009
Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. - Hugo Pfoertner, Jul 16 2020
Also called ugly numbers, although it is not clear why. - Gus Wiseman, May 21 2021
Some woody bamboo species have extraordinarily long and stable flowering intervals that belong to this sequence. The model by Veller, Nowak & Davis justifies this observation from the evolutionary point of view. - Andrey Zabolotskiy, Jun 27 2021
Also those integers k for which, for every prime p > 5, p^(4*k) - 1 == 0 (mod 240*k). - Federico Provvedi, May 23 2022
As noted in the comments to A085152, Størmer's theorem implies that the only pairs of consecutive integers that appear as consecutive terms of this sequence are (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (15,16), (24,25), and (80,81). These all represent significant musical intervals. - Hal M. Switkay, Dec 05 2022

			From _Gus Wiseman_, May 21 2021: (Start)
The sequence of terms together with their prime indices begins:
      1: {}            25: {3,3}
      2: {1}           27: {2,2,2}
      3: {2}           30: {1,2,3}
      4: {1,1}         32: {1,1,1,1,1}
      5: {3}           36: {1,1,2,2}
      6: {1,2}         40: {1,1,1,3}
      8: {1,1,1}       45: {2,2,3}
      9: {2,2}         48: {1,1,1,1,2}
     10: {1,3}         50: {1,3,3}
     12: {1,1,2}       54: {1,2,2,2}
     15: {2,3}         60: {1,1,2,3}
     16: {1,1,1,1}     64: {1,1,1,1,1,1}
     18: {1,2,2}       72: {1,1,1,2,2}
     20: {1,1,3}       75: {2,3,3}
     24: {1,1,1,2}     80: {1,1,1,1,3}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Benoit Cloitre, Plot of abs(f(n)-s(n)) vs its mean values (blue) and vs loglog(n) (red).
M. J. Dominus, Infinite Lists in Perl.
Deborah Howard and Malcolm Longair, Harmonic Proportion and Palladio's "Quattro Libri", Journal of the Society of Architectural Historians (1982) 41 (2): 116-143.
Vaclav Kotesovec, Plot of a(n) / (exp((6*log(2)*log(3)*log(5)*n)^(1/3))/sqrt(30)) for n = 1..1200000
Rosetta Code, A collection of computer codes to compute 5-smooth numbers.
Raphael Schumacher, The Formula for the Distribution of the 3-Smooth Numbers, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv:1608.06928 [math.NT], 2016.
Sci.math, Ugly numbers.
Carl Veller, Martin A. Nowak and Charles C. Davis, Extended flowering intervals of bamboos evolved by discrete multiplication, Ecology Letters, 18 (2015), 653-659.
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Regular number.
Wikipedia, Talk:Regular number. Includes a discussion of the name.
Wikipedia, Størmer's theorem.

Subsequences: A003592, A003593, A051916 , A257997.
For p-smooth numbers with other values of p, see A003586, A002473, A051038, A080197, A080681, A080682, A080683.
Cf. A006530, A112757, A112758, A112759, A112763, A112764, A291719.
The partitions with these Heinz numbers are counted by A001399.
The conjugate opposite is A033942, counted by A004250.
The opposite is A059485, counted by A004250.
The non-3-smooth case is A080193, counted by A069905.
The conjugate is A037144, counted by A001399.
The complement is A279622, counted by A035300.
Requiring the sum of prime indices to be even gives A344297.
Cf. A000244, A002182, A002183, A035301, A056239, A112798, A261144, A344293.

import Data.Set (singleton, deleteFindMin, insert)
a051037 n = a051037_list !! (n-1)
a051037_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..500] | PrimeDivisors(n) subset [2,3,5]]; // Bruno Berselli, Sep 24 2012

A051037 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            numtheory[factorset](a) minus {2, 3,5 } ;
            if % = {} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A051037(n),n=1..100) ; # R. J. Mathar, Nov 05 2017

mx = 405; Sort@ Flatten@ Table[ 2^a*3^b*5^c, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}] (* Or *)
Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (* Robert G. Wilson v *)
With[{nn=10},Select[Union[Times@@@Flatten[Table[Tuples[{2,3,5},n],{n,0,nn}],1]],#<=2^nn&]] (* Harvey P. Dale, Feb 28 2022 *)

test(n)= {m=n; forprime(p=2,5, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,500,if(test(n),print1(n",")))

a(n)=local(m); if(n<1,0,n=a(n-1); until(if(m=n, forprime(p=2,5, while(m%p==0,m/=p)); m==1),n++); n)

list(lim)=my(v=List(),s,t); for(i=0,logint(lim\=1,5), t=5^i; for(j=0,logint(lim\t,3), s=t*3^j; while(s<=lim, listput(v,s); s<<=1))); Set(v) \\ Charles R Greathouse IV, Sep 21 2011; updated Sep 19 2016

smooth(P:vec,lim)={ my(v=List([1]),nxt=vector(#P,i,1),indx,t);
while(1, t=vecmin(vector(#P,i,v[nxt[i]]*P[i]),&indx);
if(t>lim,break); if(t>v[#v],listput(v,t)); nxt[indx]++);
Vec(v)
};
smooth([2,3,5], 1e4) \\ Charles R Greathouse IV, Dec 03 2013

is_A051037(n)=n<7||vecmax(factor(n,6)[, 1])<7 \\ M. F. Hasler, Jan 16 2015

def isok(n):
  while n & 1 == 0: n >>= 1
  while n % 3 == 0: n //= 3
  while n % 5 == 0: n //= 5
  return n == 1 #  Darío Clavijo, Dec 30 2022

from sympy import integer_log
def A051037(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(y:=x//5**i,3)[0]+1):
                c -= (y//3**j).bit_length()
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A051037gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                    heapq.heappush(h, v*p)
print(list(islice(A051037gen(), 65))) # Michael S. Branicky, Sep 17 2024

Let s(n) = Card(k | a(k)Benoit Cloitre, Dec 30 2001
The characteristic function of this sequence is given by:
  Sum_{n>=1} x^a(n) = Sum_{n>=1} -Möbius(30*n)*x^n/(1-x^n). - Paul D. Hanna, Sep 18 2011
a(n) = A143207(n) / 30. - Reinhard Zumkeller, Sep 13 2011
A204455(15*a(n)) = 15, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
A006530(a(n)) <= 5. - Reinhard Zumkeller, May 16 2015
Sum_{n>=1} 1/a(n) = Product_{primes p <= 5} p/(p-1) = (2*3*5)/(1*2*4) = 15/4. - Amiram Eldar, Sep 22 2020

The "odd part" (largest odd divisor) of the number of divisors of n is a function of the exponents >=2 in the prime factorization of n (cf. A212172, A212181).

The number 1 appears a total of 18 times (see Graeme link for proof). Ramanujan proved that no number appears an infinite number of times (see Ramanujan link). It would be interesting to know more about a) which odd numbers appear in the sequence and b) how many times a number of a given size can appear in the sequence. See also A160233.

Complement of A236562. - Jaroslav Krizek, Feb 09 2014
Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - Antti Karttunen, Oct 06 2015
Since A000005(x) <= 1 + x/2, k is in the sequence if there are no x <= 2*(k+1) with k = x - d(x). - Robert Israel, Oct 12 2015
This can be improved as: k is in the sequence if there are no x <= k + A002183(2+A261100(k)) with k = x - d(x). Cf. also A070319, A262686. - Antti Karttunen, Oct 12 2015
Luca (2005) proved that this seqeunce is infinite. - Amiram Eldar, Jul 26 2025

cp's OEIS Frontend

A243220 Number of divisors of A002183(n).

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A160233 Numbers k such that k is a member of A002183 but 2*k is not.

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A212183 Largest odd divisor of A002183(n) (number of divisors of n-th highly composite number).

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

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A002182 Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.

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A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.

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