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Search: a083917 -id:a083917
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A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
(Formerly M0246 N0086) 		+10
4876
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
REFERENCES
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 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.
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).
LINKS
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.
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
FORMULA
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)
EXAMPLE
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 + ...
MAPLE
with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = if( n==0, 0, numdiv(n))}; /* Michael Somos, Apr 27 2003 */
(PARI) {a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* Michael Somos, Apr 27 2003 */
(PARI) {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
(Magma) [ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(MuPAD) numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008
(Sage) [sigma(n, 0) for n in range(1, 105)] # Zerinvary Lajos, Jun 04 2009
(Haskell)
divisors 1 = [1]
divisors n = (1:filter ((==0) . rem n)
[2..n `div` 2]) ++ [n]
a = length . divisors
-- James Spahlinger, Oct 07 2012
(Haskell)
a000005 = product . map (+ 1) . a124010_row -- Reinhard Zumkeller, Jul 12 2013
(Python)
from sympy import divisor_count
for n in range(1, 20): print(divisor_count(n), end=', ') # Stefano Spezia, Nov 05 2018
(GAP) List([1..150], n->Tau(n)); # Muniru A Asiru, Mar 05 2019
(Julia)
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
CROSSREFS
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, A005237, A005238, A006601, A006558, A019273, A039665, A049051, A001826, A001842, A049820, A051731, A066446, A106737, A129510, A115361, A129372, A127093, A143319, A061017, A091202, A091220, A156552, A159933, A159934, A027750, A163280, A183063, A263730, A034296, A237665.
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.
Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).
KEYWORD
easy,core,nonn,nice,mult,hear
AUTHOR
N. J. A. Sloane
EXTENSIONS
Incorrect formula deleted by Ridouane Oudra, Oct 28 2021
STATUS
approved
A083910 Number of divisors of n that are congruent to 0 modulo 10. +10
12
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,20
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - Franklin T. Adams-Watters, Apr 15 2007
G.f.: Sum_{k>=1} x^(10*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = (2*gamma - 1 - log(10))/10 = -0.214815..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
ndc10[n_]:=Count[Divisors[n], _?(Divisible[#, 10]&)]; Array[ndc10, 110] (* Harvey P. Dale, Jan 05 2013 *)
a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(Haskell)
a083910 = sum . map (a000007 . a010879) . a027750_row
-- Reinhard Zumkeller, Jan 15 2013
(PARI) a(n)=if(n%10, 0, numdiv(n/10)) \\ Charles R Greathouse IV, Sep 27 2015
CROSSREFS
Cf. A000005, A000007, A001227, A010879, A027750, A183063.
Cf. A001620, A002392.
Cf. A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083911 Number of divisors of n that are congruent to 1 modulo 10. +10
12
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,11
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^k/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,10) - (1 - gamma)/10 = 0.769838..., gamma(1,6) = -(psi(1/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 1 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 1); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A306716 (psi(1/10)).
Cf. A083910, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083913 Number of divisors of n that are congruent to 3 modulo 10. +10
12
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,33
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,10) - (1 - gamma)/10 = 0.07771547..., gamma(3,10) = -(psi(3/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 3 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 3); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A354642 (psi(3/10)).
Cf. A083910, A083911, A083912, A083914, A083915, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083919 Number of divisors of n that are congruent to 9 modulo 10. +10
12
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,99
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n).
G.f.: Sum_{k>=1} x^(9*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(9,10) - (1 - gamma)/10 = -0.197044..., gamma(9,10) = -(psi(9/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 9 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 9); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A354644 (psi(9/10)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083912 Number of divisors of n that are congruent to 2 modulo 10. +10
11
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,10) - (1 - gamma)/10 = 0.256367..., gamma(2,10) = -(psi(1/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
a[n_] := Sum[If[Mod[d, 10] == 2, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) A083912(n) = sumdiv(n, d, 2==(d%10)); \\ Antti Karttunen, Jan 22 2020
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A200135 (psi(1/5)).
Cf. A083910, A083911, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083914 Number of divisors of n that are congruent to 4 modulo 10. +10
11
0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,24
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n =  1..100000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(4*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,10) - (1 - gamma)/10 = -0.0163984..., gamma(4,10) = -(psi(2/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
Table[Count[Divisors[n], _?(Mod[#, 10]==4&)], {n, 110}] (* Harvey P. Dale, Dec 09 2014 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 4 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) A083914(n) = sumdiv(n, d, (4==(d%10))); \\ Antti Karttunen, Nov 07 2018
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A200136 (psi(2/5)).
Cf. A083910, A083911, A083912, A083913, A083915, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083915 Number of divisors of n that are congruent to 5 modulo 10. +10
11
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,15
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,10) - (1 - gamma)/10 = 0.0761859..., gamma(5,10) = -(psi(1/2) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
Table[Count[Divisors[n], _?(Mod[#, 10]==5&)], {n, 120}] (* Harvey P. Dale, Jan 26 2018 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 5 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 5); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A020759 (psi(1/2)).
Cf. A083910, A083911, A083912, A083913, A083914, A083916, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083916 Number of divisors of n that are congruent to 6 modulo 10. +10
11
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,36
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(6*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(6,10) - (1 - gamma)/10 = -0.118475..., gamma(6,10) = -(psi(3/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
Table[Count[Divisors[n], _?(Mod[#, 10]==6&)], {n, 110}] (* Harvey P. Dale, Oct 16 2013 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 6 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 6); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A200137 (psi(3/5)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083917, A083918, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
A083918 Number of divisors of n that are congruent to 8 modulo 10. +10
11
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,48
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
FORMULA
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083919(n).
G.f.: Sum_{k>=1} x^(8*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(8,10) - (1 - gamma)/10 = -0.176036..., gamma(8,10) = -(psi(4/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023
MATHEMATICA
Table[Count[Divisors[n], _?(Mod[#, 10]==8&)], {n, 110}] (* Harvey P. Dale, Sep 28 2016 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 8 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, d % 10 == 8); \\ Amiram Eldar, Dec 30 2023
CROSSREFS
Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A200138 (psi(4/5)).
Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083919.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, May 08 2003
STATUS
approved
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