A098198 -id:A098198 - 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

A002088 Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

Offset: 0

N. J. A. Sloane

Number of elements in the set {(x,y): 1 <= x <= y <= n, 1=gcd(x,y)}. - Michael Somos, Jun 13 1999
Sum_{k=1..n} phi(k) gives the number of distinct arithmetic progressions which contain an infinite number of primes and whose difference does not exceed n. E.g., {1k+1}, {2k+1}, {3k+1, 3k+2}, {4k+1, 4k+3}, {5k+1, ..5k+4} means 10 sequences. - Labos Elemer, May 02 2001
The quotient A024916(n)/a(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 = zeta(2)^2 = A098198 ~2.705808084277845. - Labos Elemer, Sep 20 2004 (corrected by Peter Pein, Apr 28 2009)
Also the number of rationals p/q in (0,1] with denominators q<=n. - Franz Vrabec, Jan 29 2005
a(n) is the number of initial segments of Beatty sequences for real numbers > 1, cut off when the next term in the sequence would be >= n. For example, the sequence 1,2 is included for n=3 and n=4, but not for n >= 5 because the next term of the Beatty sequence must be 3 or 4. Problem suggested by David W. Wilson. - Franklin T. Adams-Watters, Oct 19 2006
Number of complex numbers satisfying any one of {x^1=1, x^2=1, x^3=1, x^4=1, x^5=1, ..., x^n=1}. - Paul Smith (math.idiot(AT)gmail.com), Mar 19 2007
a(n+2) equals the number of Sturmian words of length n which are 'special', prefix of two Sturmian words of length n+1. - Fred Lunnon, Sep 05 2010
For n > 1: A020652(a(n)) = 1 and A038567(a(n)) = n; for n > 0: A214803(a(n)) = 1. - Reinhard Zumkeller, Jul 29 2012
Also number of elements in the set {(x,y): 1 <= x + y <= n, x >= 0, y > 0, with x and y relatively prime integers}. Thus, the number of reduced rational numbers x/y with x nonnegative, y positive, and x + y <= n. (For n >= 1, 0 <= x/y <= n - 1, clearly including each integer in this interval.) - Rick L. Shepherd, Apr 08 2014
This function, the partial sums of phi = A000010, is sometimes denoted by (uppercase) Phi. - M. F. Hasler, Apr 18 2015
From Roger Ford, Jan 16 2016: (Start)
For n >= 1: a(n) is the number of perfect arched semi-meander solutions with n arches. To be perfect the number of arch groupings must equal the number of arches with a length of 1 in the current generation and every preceding generation.
Example:  p is the number of arches with length 1 (/\),  g is the number of arch groups (-),  n is number of arches in the top half of a semi-meander solution
                 /\
        /\      //\\
       //\\-/\-///\\\-  n=6 p=3 g=3   Each preceding arch configuration
            /\   /\                   is formed by attaching the arch
        /\-//\\-//\\-   n=5 p=3 g=3   end in the first position and the
           /\                         arch end in the last position.
          //\\
         ///\\\-/\-     n=4 p=2 g=2
           /\
          //\\-/\-      n=3 p=2 g=2
           /\-/\-       n=2 p=2 g=2
            /\-         n=1 p=1 g=1. (End)
a(n) is the number of distinct lists of binary words of length n that are balanced (Sturmian). - Dan Rockwell, Will Wodrich, Aaliyah Fiala, and Bob Burton, May 30 2019
2013 IMO Problem 6 shows that a(n) is the number of ways to arrange the numbers 0, 1, ..., n on a circle such that for any numbers 0 <= a < b < c < d <= n, the chord joining a and d does not intersect with the chord intersecting b and c, with rotation counted as same. - Yifan Xie, Aug 26 2025

			G.f. = x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 12*x^6 + 18*x^7 + 22*x^8 + 28*x^9 + ...

A. Beiler, Recreations in the Theory of Numbers, Dover Publications, 1966, Chap. XVI.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 138.
M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 6.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section I.21.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 94, Problem 11.
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).
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 111.

Al Zimmermann, Table of n, a(n) for n = 0..50000 (Terms 0 to 1000 by T. D. Noe)
Dorin Andrica, Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Art of Problem Solving, 2013 IMO Problem 6.
S. Bockting-Conrad, Y. Kashina, T. K. Petersen, and B. E. Tenner, Sós permutations, arXiv:2007.01132 [math.CO], 2020.
Rayan Chikhi, Vladan Jovicic, Stefan Kratsch, Paul Medvedev, Martin Milanic, Sofya Raskhodnikova, and Nithin Varma, Bipartite Graphs of Small Readability, arXiv:1805.04765 [cs.DM], 2018.
Steven R. Finch, Euler Totient Function Asymptotic Constants [Broken link]
Steven R. Finch, Euler Totient Function Asymptotic Constants [From the Wayback machine]
Paul Loomis, Michael Plytage and John Polhill, Summing up the Euler phi function, The College Mathematics Journal, Vol. 39, No. 1, Jan. 2008, pp. 34-42.
J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, p. 24.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
James J. Sylvester, On the number of fractions contained in any Farey series of which the limiting number is given, in: London, Edinburgh and Dublin Philosophical Magazine (5th series) 15 (1883), p. 251 = Collected Mathematical Papers, Vols. 1-4, Cambridge Univ. Press, 1904-1912, Vol. 4, p. 103 (see below).
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin 1963.
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Beatty Sequence.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
R. G. Wilson, v, Letter to N. J. A. Sloane, Jan 24 1989

Cf. A000010, A015614, A005728, A067282, A001088, A134542, A002321.

List([1..60],n->Sum([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018

a002088 n = a002088_list !! n
a002088_list = scanl (+) 0 a000010_list -- Reinhard Zumkeller, Jul 29 2012

[&+[EulerPhi(i): i in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Aug 01 2018

with(numtheory): A002088:=n->add(phi(i),i=1..n): seq(A002088(n), n=0..70);

Table[Plus @@ EulerPhi[Range[n]], {n, 0, 57}] (* Alonso del Arte, May 30 2006 *)
Accumulate[EulerPhi[Range[0,60]]] (* Harvey P. Dale, Aug 27 2011 *)

a(n)=sum(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Jun 16 2011

a(n)=my(s=1); forsquarefree(k=1,n,s+=(n\k[1])^2*moebius(k)); s/2 \\ Charles R Greathouse IV, Oct 15 2021

first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]]=s+=eulerphi(k)); v \\ Charles R Greathouse IV, Oct 15 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A002088(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A002088(k1)-1)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 24 2021

[sum(euler_phi(k) for k in (1..n)) for n in (0..60)] # G. C. Greubel, Nov 25 2018

a(n) = (3*n^2)/(Pi^2) + O(n log n).
More precisely, a(n) = (3/Pi^2)*n^2 + O(n*(log(n))^(2/3)*(log(log(n)))^(4/3)), (A. Walfisz 1963). - Benoit Cloitre, Feb 02 2003
a(n) = (1/2)*Sum_{k>=1} mu(k)*floor(n/k)*floor(1+n/k). - Benoit Cloitre, Apr 11 2003
a(n) = A000217(n) - A063985(n) = A018805(n) - A015614(n). - Reinhard Zumkeller, Jan 21 2013
A slightly simpler version of Cloitre's formula is a(n) = 1/2 + Sum_{k=1..oo} floor(n/k)^2*mu(k)/2. - Bill Gosper, Jul 25 2020
The quotient A024916(n)/a(n) = SummatorySigma/SummatoryTotient as n increases seems to approach (Pi^4)/36 = Zeta(2)^2 = 2.705808084277845. See also A067282. - Labos Elemer, Sep 21 2004
A024916(n)/a(n) = zeta(2)^2 + O(log(n)/n). This follows from asymptotic formulas for the sequences. - Franklin T. Adams-Watters, Oct 19 2006
Row sums of triangle A134542. - Gary W. Adamson, Oct 31 2007
G.f.: (Sum_{n>=1} mu(n)*x^n/(1-x^n)^2)/(1-x), where mu(n) = A008683(n). - Mamuka Jibladze, Apr 06 2015
a(n) = A005728(n) - 1, for n >= 0. - Wolfdieter Lang, Nov 22 2016
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (M(floor(n/k)) - M(floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k) * floor(n/k) * floor(1+n/k))/2, where M(k) is the Mertens function (A002321) and mu(k) is the Moebius function (A008683). - Daniel Suteu, Nov 23 2018
a(n) = A015614(n)+1. - R. J. Mathar, Apr 26 2023
a(n) = A000217(n) - Sum{k=2..n} a(floor(n/k)). From summing over Id = 1 (Dirichlet convolution) phi. - Jason Xu, Jul 31 2024
a(n) = Sum_{k=1..n} k*A002321(floor(n/k)). - Ridouane Oudra, Jul 03 2025

Additional comments from Len Smiley

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

Original entry on oeis.org

2, 8, 6, 7, 4, 7, 4, 2, 8, 4, 3, 4, 4, 7, 8, 7, 3, 4, 1, 0, 7, 8, 9, 2, 7, 1, 2, 7, 8, 9, 8, 3, 8, 4, 4, 6, 4, 3, 4, 3, 3, 1, 8, 4, 4, 0, 9, 7, 0, 5, 6, 9, 9, 5, 6, 4, 1, 4, 7, 7, 8, 5, 9, 3, 3, 6, 6, 5, 2, 2, 4, 3, 1, 3, 1, 9, 4, 3, 2, 5, 8, 2, 4, 8, 9, 1, 2, 6, 8, 2, 5, 5, 3, 7, 4, 2, 3, 7, 4, 6, 8, 5, 3, 6, 4, 7

Offset: 0

N. J. A. Sloane, Nov 19 2001

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

			0.2867474284344787341078927127898384...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 11 2016 *)

prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 17 2021

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016
The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - Amiram Eldar, May 24 2020
Equals A065472 * A227929 =  A065472 / A098198. - Amiram Eldar, Aug 04 2020

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019

A152649 Decimal expansion of Pi^4/72.

Original entry on oeis.org

1, 3, 5, 2, 9, 0, 4, 0, 4, 2, 1, 3, 8, 9, 2, 2, 7, 3, 9, 3, 9, 5, 0, 0, 4, 6, 2, 0, 6, 7, 6, 4, 5, 9, 8, 7, 8, 4, 6, 8, 4, 3, 8, 6, 8, 9, 8, 9, 8, 4, 0, 8, 6, 3, 4, 6, 0, 3, 7, 2, 0, 2, 6, 9, 3, 0, 5, 1, 5, 0, 7, 7, 0, 2, 3, 3, 7, 1, 1, 0, 5, 8, 1, 9, 6, 1, 3, 7, 0, 4, 4, 9, 2, 7, 1, 2, 4, 8, 9, 6, 5, 4, 1, 2, 3

Offset: 1

R. J. Mathar, Dec 10 2008

A division by 2 is missing in Mezo's penultimate formula on page 4.

			Equals 1.352904042138922739395004620676459878468438689898408634603...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995), 1191-1198.
I. Gradsteyn and I. Ryzhik, Table of integrals, series and products, Academic Press, 1980, page 7 (formulas from 0.233.3 to 0.233.5).
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.
Index entries for transcendental numbers.

Cf. A001008, A002805, A048272, A092425, A098198.

evalf(Pi^4/72,120); # Muniru A Asiru, Sep 18 2018

RealDigits[Pi^4/72,10,120][[1]] (* Harvey P. Dale, Feb 10 2013 *)

PARI
```
Pi^4/72 \\ Michel Marcus, Jul 07 2015
```

Equals A098198/2 = A092425/72.
Equals Sum_{j >= 1} H(j)/j^3 where H(j) = A001008(j)/A002805(j).
Equals 20*Sum_{j >= 1} (2*j)^(-4) (see Gradsteyn and Ryzhik in Links section). - A.H.M. Smeets, Sep 18 2018
Equals Sum_{k>=1} A048272(k)/k^2. - Amiram Eldar, Jan 25 2024

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Dec 26 2023

The odd terms of A005361.

The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

a(n) = A005361(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A333972 Decimal expansion of Pi^6/540 = zeta(2) * zeta(4).

Original entry on oeis.org

1, 7, 8, 0, 3, 5, 0, 3, 5, 8, 4, 7, 2, 7, 8, 5, 9, 9, 4, 5, 0, 0, 4, 0, 6, 3, 7, 7, 1, 3, 4, 1, 1, 0, 9, 2, 3, 8, 2, 8, 1, 8, 0, 6, 0, 7, 5, 5, 7, 4, 9, 3, 7, 3, 3, 2, 2, 4, 2, 1, 5, 1, 6, 2, 0, 0, 7, 5, 8, 1, 3, 2, 0, 0, 7, 8, 4, 2, 6, 3, 2, 1, 2, 9, 4, 8, 5, 4, 4, 6, 1, 3, 9, 2, 4

Offset: 1

Bernard Schott, Sep 29 2020

Compare 1st formula with Sum_{m>0, q>0} 1/(m^2*q^2) = Pi^4/36 = (zeta(2))^2 = A098198.

			1.78035035847278599450040637713411092382818060755749373322421516...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.22, p. 275.

Index entries for transcendental numbers

Cf. A001157, A013661, A046951, A068447, A098198.

Maple
```
evalf(Pi^6/540,120);
```

RealDigits[Pi^6/540, 10, 100][[1]] (* Amiram Eldar, Sep 29 2020 *)

PARI
```
Pi^6/540 \\ Michel Marcus, Sep 30 2020
```

Equals Sum_{m>0, q>0, m | q} 1/(m^2*q^2).
Equals A013661 * A068447.
Equals Sum_{k>=1} sigma_2(k)/k^4. - Amiram Eldar, Sep 30 2020
Equals Sum_{k>=1} A046951(k)/k^2. - Amiram Eldar, Jan 25 2024

A371415 Dedekind psi function applied to the cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 12, 36, 48, 150, 192, 432, 324, 392, 768, 1728, 1800, 1452, 3888, 3072, 2916, 2366, 4704, 3750, 5400, 6912, 7200, 5202, 7220, 15552, 12288, 14112, 17424, 18816, 12696, 27648, 28800, 19208, 34992, 28392, 26244, 25230, 45000, 64800, 30752, 48600, 62208, 49152

Offset: 1

Amiram Eldar, Mar 22 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001615, A065464, A098198, A335988.

Similar sequences: A323332, A371413, A371414.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[40000], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f, ans); print1(1, ", "); for(k = 2, max, f = factor(k); ans = 1; for (i = 1, #f~, if (f[i, 2] == 1 || !(f[i, 2] % 2), ans = 0; break)); if(ans, print1(dedpsi(f), ", ")));}

a(n) = A001615(A335988(n)).

Sum_{n>=1} 1/a(n) = (Pi^4/36) * Product_{p prime} (1 - (2*p-1)/p^3) = A098198 * A065464 = 1.158760974549073218921828... .

A183030 Decimal expansion of Sum_{j>=1} tau(j)/j^3 = zeta(3)^2.

Original entry on oeis.org

1, 4, 4, 4, 9, 4, 0, 7, 9, 8, 4, 3, 3, 6, 3, 4, 2, 3, 3, 9, 1, 3, 6, 8, 5, 0, 7, 8, 8, 0, 6, 9, 8, 7, 8, 2, 7, 1, 8, 3, 7, 3, 5, 4, 0, 5, 7, 6, 3, 8, 8, 8, 6, 7, 4, 1, 3, 1, 4, 3, 4, 1, 6, 1, 8, 9, 8, 5, 8, 3, 8, 5, 6, 1, 3, 1, 3, 5, 4, 1, 0, 1, 9, 6, 6, 1, 9

Offset: 1

R. J. Mathar, Dec 18 2010

This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=3. At s=2 we find A098198, at s=4 A183031.

Since tau(n)/n^3 is a multiplicative function, one finds an Euler product for the sum, which is expanded with an Euler transformation to a product of Riemann zeta functions as in A175639 for numerical evaluation.

			1.4449407984336342339136.. = 1 +2/2^3 +2/3^3 +3/4^3 +2/5^3 +4/6^3 +2/7^3+...

Cf. A000005, A002117, A098198, A175639, A183031.

Maple
```
evalf(Zeta(3)^2);
```

RealDigits[Zeta[3]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

zeta(3)^2 \\ Charles R Greathouse IV, Mar 04 2015

Equals the Euler product Product_{p= A000040} (1+ (2*p^s-1)/(p^s-1)^2) at s=3, or the square of A002117.

A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

Original entry on oeis.org

0, 2, 2, 7, 2, 14, 2, 18, 9, 18, 2, 43, 2, 22, 20, 41, 2, 54, 2, 57, 24, 30, 2, 106, 13, 34, 31, 71, 2, 110, 2, 88, 32, 42, 28, 162, 2, 46, 36, 142, 2, 138, 2, 99, 81, 54, 2, 237, 17, 102, 44, 113, 2, 178, 36, 178, 48, 66, 2, 325, 2, 70, 99, 183, 40, 194, 2

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Numbers n such that n divides a(n) are given in A068978.

			For n = 12: Sum_{d|n, d<n} d * tau(n / d) = 1*6 + 2*4 + 3*3 + 4*2 + 6*2 = 43.

Antti Karttunen, Table of n, a(n) for n = 1..27144 (first 1000 terms from Jaroslav Krizek)

Cf. A000005, A000203, A007429, A068978, A098198, A211779.

Table[Sum[d*DivisorSigma[0, n/d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 27 2012 *)

A211780(n) = sumdiv(n, d, sigma(d))-n; \\ Antti Karttunen, Nov 13 2017

A211780=lambda n:sum(sigma(d) for d in divisors(n, generator=True))-n
from sympy import divisor_sigma as sigma, divisors # M. F. Hasler, Jun 03 2024

a(n) = A007429(n) - n = A211779(n) + A000203(n) - n .
a(n) = (Sum_{d|n} A000203(d)) - n. - Antti Karttunen, Nov 13 2017
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^4/36 - 1 = 1.705808... . - Amiram Eldar, Jun 06 2024

Name edited by M. F. Hasler, Jun 03 2024

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 2]+1));}

Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.
a(n) = A000005(A368167(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A000005(n), with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 3/p^3 - 2/p^4 - 1/p^5 + 1/p^6) = 1.47140789970892803631... .

cp's OEIS Frontend

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

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A002088 Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.

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A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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A152649 Decimal expansion of Pi^4/72.

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Formula

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

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