A005238 -id:A005238 - 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

A005237 Numbers k such that k and k+1 have the same number of divisors.

Original entry on oeis.org

2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 104, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 189, 201, 202, 205, 213, 214, 217, 218, 230, 231, 242, 243, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 344, 374, 375, 381, 387

Offset: 1

N. J. A. Sloane

nonn

Is a(n) asymptotic to c*n with 9 < c < 10? - Benoit Cloitre, Sep 07 2002
Let S = {(n, a(n)): n is a positive integer < 2*10^5}, where {a(n)} is the above sequence. The best-fit (least squares) line through S has equation y = 9.63976*x - 1453.76. S is very linear: the square of the correlation coefficient of {n} and {a(n)} is about 0.999943. - Joseph L. Pe, May 15 2003
I conjecture the contrary: the sequence is superlinear. Perhaps a(n) ~ n log log n. - Charles R Greathouse IV, Aug 17 2011
Erdős proved that this sequence is superlinear. Is a more specific result known? - Charles R Greathouse IV, Dec 05 2012
Heath-Brown proved that this sequence is infinite. Hildebrand and Erdős, Pomerance, & Sárközy show that n sqrt(log log n) << a(n) << n (log log n)^3, where << is Vinogradov notation. - Charles R Greathouse IV, Oct 20 2013

			14 is in the sequence because 14 and 15 are both in A030513. 104 is in the sequence because 104 and 105 are both in A030626.  - _R. J. Mathar_, Jan 09 2022

R. K. Guy, Unsolved Problems in Number Theory, B18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 840.
P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530-540.
P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), pp. 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129:2 (1987), pp. 307-319.

Cf. A000005, A005238, A006601, A049051, A006558, A019273, A039665, A073915.

Equals A083795(n-1) - 1.

A005237Q = DivisorSigma[0, #] == DivisorSigma[0, # + 1] &; Select[Range[387], A005237Q] (* JungHwan Min, Mar 02 2017 *)
SequencePosition[DivisorSigma[0,Range[400]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 25 2019 *)

is(n)=numdiv(n)==numdiv(n+1) \\ Charles R Greathouse IV, Aug 17 2011

from sympy import divisor_count as tau
[n for n in range(1,401) if tau(n) == tau(n+1)] # Karl V. Keller, Jr., Jul 10 2020

More terms from Jud McCranie, Oct 15 1997

A006558 Start of first run of n consecutive integers with same number of divisors.

Original entry on oeis.org

1, 2, 33, 242, 11605, 28374, 171893, 1043710445721, 2197379769820, 2642166652554075

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The entry 40311 given by Guy and by Wells is incorrect. - Jud McCranie, Jan 20 2002
a(10) <= 2642166652554075, a(11) <= 17707503256664346, a(12) <= 9827470582657267545. - David Wasserman, Feb 22 2008
a(10) > 10^13. - Giovanni Resta, Jul 13 2015
a(12) <= 3842083249515874843. - Hugo van der Sanden, Sep 20 2022
a(13) <= 34169215324203592637988571. - Hugo van der Sanden, Apr 13 2022
a(14) <= 9721439902882994590514319997146. - Hugo van der Sanden, Jun 14 2022
a(15) <= 80215613469168729088982885848674841. - Natalia Makarova, Sep 18 2022
a(16) <= 37981337212463143311694743672867136611416. - Vladimir Letsko, Mar 17 2017
a(17) <= 768369049267672356024049141254832375543516. - Vladimir Letsko, Sep 12 2017
a(18) <= 488900003598703704335810037459507226590256411. - Vladimir Letsko, Jun 03 2022
a(19) <= 5908388043825578351730345292813071711296723319324. - Vladimir Letsko, Apr 09 2022
a(20) <= 17668887847524548413038893976018715843277693308027547. Vladimir Letsko, May 30 2022
Spătaru proves that the longest such run up to N is at most exp(C*sqrt(log N log log N)) for some constant C, hence a(n) >> exp(exp(W((log^2 n)/C))) which is approximately exp(log^2 n/(2 log log n)). - Charles R Greathouse IV, Feb 06 2023

			33 has four divisors (1, 3, 11, and 33), 34 has four divisors (1, 2, 17, and 34), 35 has four divisors (1, 5, 7, and 35).  These are the first three consecutive numbers with the same number of divisors, so a(3)=33.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, section B18.
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 87.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, pages 147 and 176.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 Nos. 1-4 (1993) pp. 166-168. See p. 167.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko and Vasilii Dziubenko On consecutive equidivisible integers (in Russian), Boundaries of knowledge, 2 (45) 2016.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 61: problem 20 revisited, The Prime Puzzles and Problems Connection.
Vlad-Titus Spătaru, Runs of consecutive integers having the same number of divisors, arXiv preprint (2023). arXiv:2301.04464 [math.NT]

Cf. A000005, A005237, A005238, A006601, A049051, A019273, A039665.

Cf. A034173, A115158, A119479.

tau = DivisorSigma[0, #]&;
A006558[q_, w_] := Module[{a, k, j, ok, n}, For[j = 0, j <= w, j++, For[n = 1, n <= q, n++, ok = 1; a = tau[n]; For[k = 1, k <= j, k++, If[a != tau[n + k], ok = 0; Break[]]]; If [ok == 1, Print[n]; Break[]]]]];
A006558[2*10^5, 7] (* Jean-François Alcover, Dec 10 2017 *)

isok(n, k)=nb = numdiv(k); for (j=k+1, k+n-1, if (numdiv(j) != nb, return(0));); 1;
a(n) = {k=1; while (!isok(n, k), k++); k;} \\ Michel Marcus, Feb 17 2016

a(8) from Jud McCranie, Jan 20 2002
a(9) conjectured by David Wasserman, Jan 08 2006
a(9) confirmed by Jud McCranie, Jan 14 2006
a(10) by Jud McCranie, Nov 27 2018

A006601 Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors.

Original entry on oeis.org

242, 3655, 4503, 5943, 6853, 7256, 8392, 9367, 10983, 11605, 11606, 12565, 12855, 12856, 12872, 13255, 13782, 13783, 14312, 16133, 17095, 18469, 19045, 19142, 19143, 19940, 20165, 20965, 21368, 21494, 21495, 21512, 22855, 23989, 26885

Offset: 1

N. J. A. Sloane

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.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B18, pp. 111-113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
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].
Victor Meally, Letter to N. J. A. Sloane, no date.

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A049051 (runs of 5), A049052 (runs of 6), A049053 (runs of 7).

import Data.List (elemIndices)
a006601 n = a006601_list !! (n-1)
a006601_list = map (+ 1) $ elemIndices 0 $
   zipWith3 (((+) .) . (+)) ds (tail ds) (drop 2 ds) where
   ds = map abs $ zipWith (-) (tail a000005_list) a000005_list
-- Reinhard Zumkeller, Jan 18 2014

f[n_]:=Length[Divisors[n]]; lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3],AppendTo[lst,n]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
dsQ[n_]:=Length[Union[DivisorSigma[0,Range[n,n+3]]]]==1; Select[Range[ 30000],dsQ] (* Harvey P. Dale, Nov 23 2011 *)
Flatten[Position[Partition[DivisorSigma[0,Range[27000]],4,1],?(Union[ Differences[ #]]=={0}&),{1},Heads->False]] (* Faster, because the number of divisors for each number is only calculated once *) (* _Harvey P. Dale, Nov 06 2013 *)
SequencePosition[DivisorSigma[0,Range[27000]],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 03 2017 *)

is(n)=my(t=numdiv(n)); numdiv(n+1)==t && numdiv(n+2)==t && numdiv(n+3)==t \\ Charles R Greathouse IV, Jun 25 2017

More terms from Olivier Gérard

A049051 Numbers k such that k through k+4 all have the same number of divisors.

Original entry on oeis.org

11605, 12855, 13782, 19142, 21494, 28374, 28375, 40311, 42805, 50582, 55254, 60231, 60663, 79094, 87655, 90181, 90182, 95845, 99655, 103621, 109765, 115591, 120727, 121045, 122151, 122871, 142454, 142806, 152630, 157493, 157494, 171893

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A000005, A006558, A019273, A039665, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A006601 (runs of 4), A049052 (runs of 6), A049053 (runs of 7).

Flatten[Position[Partition[DivisorSigma[0,Range[200000]],5,1],?(Length[ Union[#]] == 1&),{1},Heads->False]] (* _Harvey P. Dale, May 30 2014 *)
SequencePosition[DivisorSigma[0,Range[200000]],{x_,x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 03 2017 *)

A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

Original entry on oeis.org

1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, 6853, 6985, 7257, 7953, 8393, 8533, 8785, 9213, 9453, 9821, 9877, 10281, 10945, 11605, 12453, 12565, 12801, 12857, 12993, 13053, 14133, 14313, 14329, 14465, 14817, 15085, 15265, 15805, 16113, 16133

Offset: 1

Jason Earls, Jan 04 2002

nonn

A subsequence of A052214 and thus of A005238. - M. F. Hasler, Jan 05 2013
Also, the start of pairs of adjacent sphenic twins, i.e., a(n) = A215217(k) such that A215217(k+1) = A215217(k)+1. Therefore these triples might be called "sphenic triples". They form a subsequence of A242606. - M. F. Hasler, May 18 2014
Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - Zak Seidov, Jul 04 2020

			a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 1309.

Subsequence of A052214 and hence of A005238.

Subsequence of A215217, A007675, A242606 and A168626.

f[n_]:=Last/@FactorInteger[n]=={1,1,1};lst={};Do[If[f[n]&&f[n+1]&&f[n+2],AppendTo[lst,n]],{n,9!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 04 2010 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,17000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 28 2025 *)

Trip(n) = { local(f); f=factor(n); if (matsize(f)[1] != 3, return(0)); for(i=1, 3, if (f[i, 2] != 1, return(0))); return(1); } { n=0; for (m=1, 10^10, if (!Trip(m) || !Trip(m+1) || !Trip(m+2), next); write("b066509.txt", n++, " ", m); if (n==1000, return) ) } \\ Harry J. Smith, Feb 19 2010

A066509(n,show_all=0,a=2*3*5,s=[1,1,1]~)={until( !n-- || !a++, until(, factor(a+2)[,2]!=s && (a+=3) && next; factor(a+1)[,2]!=s && (a+=2) && next; factor(a)[,2]==s && break; factor(a+3)[,2]==s && a++ && break; a+=4);show_all && print1(a",")); a} \\ M. F. Hasler, Jan 05 2013

is3dp(n)=my(f=factor(n));matsize(f)==[3,2]&&vecmax(f[,2])==1
list(lim)=my(v=List(),t);forprime(p=17,lim\15, forprime(q=5,min(p-1,lim\3), forprime(r=3,min(q-1,lim\(p*q)), t=p*q*r; if(t%4==1 && is3dp(t+1) && is3dp(t+2), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Jan 05 2013; updated Jan 22 2025

list(lim)=my(v=List(),ct); forfactored(n=1309,lim\1+2, if(n[2][,2]==[1,1,1]~, if(ct++==3, listput(v,n[1]-2)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Aug 30 2022

a(n) == 1 (mod 4). - Zak Seidov, Mar 31 2020

Definition clarified by Harvey P. Dale, Feb 28 2025

A052214 Numbers n with prime signature(n) = prime signature(n+1) = prime signature(n+2).

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 603, 633, 697, 921, 1041, 1137, 1261, 1274, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1924, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2523, 2641, 2665, 2721, 2733, 3097, 3385

Offset: 1

Erich Friedman, Jan 29 2000

			33 = 3^1 11^1, 34 = 2^1 17^1 and 35 = 5^1 7^1, so all have prime signature {1,1}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A subsequence of A005238. - M. F. Hasler, Jan 05 2013

Cf. A075039.

pri[ n_ ] := Sort[ Transpose[ FactorInteger[ n ] ][ [ 2 ] ] ] Select[ Range[ 2,10000 ],pri[ # ]==pri[ #+1 ]==pri[ #+2 ]& ]

A052214(n, /*give optional 2nd arg =1 to print all terms*/ show_all=0, a=2*3)={until( !n-- || !a++, until(, vecsort(factor(a+=2)[, 2])!=vecsort(factor(a-1)[, 2]) && next; ((t=vecsort(factor(a-1)[, 2]))==vecsort(factor(a-2)[, 2]) || vecsort(factor(a++)[, 2])==t) && (a-=2) && break); show_all && print1(a", ")); a}  \\ - M. F. Hasler, Jan 06 2013

A019273 First run of n consecutive integers with same number of divisors ends at a(n).

Original entry on oeis.org

1, 3, 35, 245, 11609, 28379, 171899, 1043710445728, 2197379769828, 2642166652554084

Offset: 1

Jud McCranie

The entry 40315 given in Guy and Wells is incorrect.

Terminating number in the sequence of consecutive integers with the same number of divisors in A006558. - Jud McCranie, Jan 14 2006

			phi(33)=phi(34)=phi(35), so a(3)=35

R. K. Guy "Unsolved Problems in Number Theory", section B18.
David Wells "Dictionary of Curious and Interesting Numbers" #40311.

Cf. A000005, A005237, A005238, A006601, A049051, A006558, A039665.

a(n)=A006558(n)+n-1.

More terms from Jud McCranie, Oct 04 2002
One more term from Jud McCranie, Jan 14 2006
a(10) by Jud McCranie, Nov 27 2018

A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921

Offset: 1

Matthijs Coster, Aug 23 2008

Old name was "The first number of a series of 5 consecutive numbers with the same signature, i.e., all numbers have the format p^2*q, where p and q are primes. Therefore the number of divisors is the same (6)." [That name could have been confusing in that not every sequence of 5 consecutive integers having the same prime signature has the prime signature p^2*q; e.g., 204323 is the first of 5 consecutive numbers of the form p^2*q*r. - Jon E. Schoenfield, Jun 05 2018]
Each of the five numbers in each such sequence has 6 divisors.
It is easy to prove that any number in this sequence must be congruent to 1 modulo 240. The program below calculates only an element of the sequence. Since the reference A119479 it is the smallest one. If we assume that the first element has the format 7^2*n49, the second number has the format 2*p^2, the third element has the format 3^2*n9 and the fifth element has the format 5^2*n25, then p must be modulo 22050 one out of 1181, 3719, 4219, 9119, 12931, 17831, 18331 or 20869.
It is unclear if these numbers are the smallest ones. - Matthijs Coster, Aug 28 2008 [The terms listed in the Data section are, in fact, the smallest numbers matching the definition. - Jon E. Schoenfield, Jun 05 2018]
The first quintuple not of the aforementioned form starts with 5344962129269790721 = 23^2*prime. - Ivan Neretin, Feb 08 2016
Among the first 200 terms, the frequency with which the squared prime factor p is {7, 17, 23, 31, 41, 47, 73, 127, 193, 1039, 1399} is {171, 10, 6, 4, 3, 1, 1, 1, 1, 1, 1}, respectively. - Jon E. Schoenfield, Jun 09 2018

			a(1) = 10093613546512321, because
10093613546512321 = 7^2 * 205992113194129,
10093613546512322 =   2 * 71040881^2,
10093613546512323 = 3^2 * 1121512616279147,
10093613546512324 = 2^2 * 2523403386628081, and
10093613546512325 = 5^2 * 403744541860493,
so each of the five consecutive integers is of the form p^2*q, and no smaller run of five consecutive integers has this property. [corrected by _Jon E. Schoenfield_, Jun 05 2018]

Ray Chandler, Table of n, a(n) for n = 1..2000 (first 25 terms from Ivan Neretin, through 200 terms from Jon E. Schoenfield)
Richard I. Hess, Problem 1231, Crux Mathematicorum, Vol. 13, No. 4, p. 118, 1987. (takes a long time to download)
Richard I. Hess, Puzzles from Around the World, p. 63, H17.
Carlos Rivera, Problem 20.- Divisors (II) K consecutive numbers with the same number of divisors, The Prime Puzzles & Problems Connection.
StackExchange, Sequence of numbers with prime factorization pq^2
wu :: forums, Same Number of Divisors, Oct 05 2007.
Index to sequences related to prime signature

Cf. A119479, A006558, A005237, A005238, A006601.

Cf. A054753, A074172, A178032, A308683.

## Warning: this program appears to be incorrect [Joerg Arndt, Feb 29 2016]
for m in range(5000):
    p = 22050*m+17831
    if is_prime(p):
        n = 2*p^2-2
        n4 = n/4+1
        if is_prime(n4):
            n49 = floor((n+1)/49)
            if (49*n49 == n+1) and is_prime(n49):
                n9 = floor((n+3)/9)
                if (9*n9 == n+3) and is_prime(n9):
                    n25 = floor((n+5)/25)
                    if (25*n25 == n+5) and is_prime(n25):
                        print(n+1, n49, p, n9, n4, n25)

Two more terms Matthijs Coster, Aug 28 2008
Missing terms added and extended by Ivan Neretin, Feb 08 2016
New name from Jon E. Schoenfield, Jun 05 2018

cp's OEIS Frontend

A049109 Number of divisors of A005238(n).

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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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A005237 Numbers k such that k and k+1 have the same number of divisors.

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A006558 Start of first run of n consecutive integers with same number of divisors.

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A006601 Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors.

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A049051 Numbers k such that k through k+4 all have the same number of divisors.

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A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.