A211264 -id:A211264 - cp's OEIS frontend

A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, 113, 119, 123, 127, 131, 140, 142, 146, 150, 158, 160, 168, 170, 176, 182, 186, 188, 198, 201, 207, 211, 217, 219, 227, 231, 239, 243, 247, 249

Offset: 0

N. J. A. Sloane

The identity Sum_{k=1..n} floor(n/k) = Sum_{k=1..n} d(k) is Equation (10), p. 58, of Apostol (1976). - N. J. A. Sloane, Dec 06 2020
The "Dirichlet divisor problem" is to find a precise asymptotic estimate for this sequence - see formula lines below, also Apostol (1976), Chap. 3.
Number of increasing arithmetic progressions where n+1 is the second or later term. - Mambetov Timur, Takenov Nurdin, Haritonova Oksana (timus(AT)post.kg; oksanka-61(AT)mail.ru), Jun 13 2002. E.g., a(3) = 5 because there are 5 such arithmetic progressions: (1, 2, 3, 4); (2, 3, 4); (1, 4); (2, 4); (3, 4).
Binomial transform of A001659.
Area covered by overlapped partitions of n, i.e., sum of maximum values of the k-th part of a partition of n into k parts. - Jon Perry, Sep 08 2005
Equals inverse Mobius transform of A116477. - Gary W. Adamson, Aug 07 2008
The Polymath project (see the Tao-Croot-Helfgott link) sketches an algorithm for computing a(n) in essentially cube root time, see section 2.1. - Charles R Greathouse IV, Oct 10 2010 [Sladkey gives another. - Charles R Greathouse IV, Oct 02 2017]
The Dirichlet inverse starts (offset 1) 1, -3, -5, 1, -10, 16, -16, 1, 2, 33, -29, -6, -37, 55, 55, -1, -52, -5, -60, ... - R. J. Mathar, Oct 17 2012
The inverse Mobius transforms yields A143356. - R. J. Mathar, Oct 17 2012
An improved approximation vs. Dirichlet is: a(n) = log(Gamma(n+1)) + 2n*gamma. Using sample ranges of {n = k^2-k to k^2 + (k-1)} the means of the new error term are < +- 0.5 up to k=150, except on two values of k. These ranges appear to give means closest to zero for such small sample sizes. It is not clear sample means remain < +- 0.5 at larger k. The standard deviations are ~(n*log(n))^(1/4)/2, with n near sample range center. - Richard R. Forberg, Jan 06 2015
The values of n for which a(n) is even are given by 4*m^2 <= n <= 4*m(m+1) for m >= 0. Example: for m=1 the values of n are 4 <= n <= 8 for which a(4) to a(8) are even. - G. C. Greubel, Sep 30 2015
For n > 0, a(n) = count(x|y), 1 <= y <= x <= n, that is, the number of pairs in the ordered list of x and y, where y divides x, up to and including n. - Torlach Rush, Jan 31 2017
a(n) is also the total number of partitions of all positive integers <= n into equal parts. - Omar E. Pol, May 29 2017
a(n) is the rank of the join of the set of elements of rank n in Young's lattice, the lattice of all integer partitions ordered by inclusion of their Ferrers diagrams. - Geoffrey Critzer, Jul 11 2018
a(n) always has the same parity as floor(sqrt(n)) = A000196(n): see A211264 (proof in Diophante link). - Bernard Schott, Feb 13 2021
From Omar E. Pol, Feb 16 2021: (Start)
Apart from initial zero this is the convolution of A341062 and A000027.
Nonzero terms convolved with A341062 gives A055507. (End)
From Bernard Schott, Apr 17 2022: (Start)
a(n-1) is the number of lattice points in the first quadrant lying under the hyperbola x*y = n, excluding the lattice points on the axes.
a(n) is the number of lattice points in the first quadrant lying on or under the hyperbola x*y = n, excluding the lattice points on the axes. (Reference Hari Kishan). (End)
Let tiles Tn (for n >= 1) be initially placed on square n on an infinite 1D board. At each step, the leftmost unblocked tile (i.e., the top tile in the leftmost stack) jumps forward exactly n squares. Tiles can stack, and only the top tile of a stack can move. This sequence gives the step number when tile n moves for the first time. - Ali Sada, May 23 2025

			a(3) = 5 because 3 + floor(3/2) + 1 = 3 + 1 + 1 = 5. Or tau(1) + tau(2) + tau(3) = 1 + 2 + 2 = 5.
a(4) = 8 because 4 + floor(4/2) + floor(4/3) + 1 = 4 + 2 + 1 + 1 = 8. Or
tau(1) + tau(2) + tau(3) + tau(4) = 1 + 2 + 2 + 3 = 8.
a(5) = 10 because 5 + floor(5/2) + floor(5/3) + floor (5/4) + 1 = 5 + 2 + 1 + 1 + 1 = 10. Or tau(1) + tau(2) + tau(3) + tau(4) + tau(5) = 1 + 2 + 2 + 3 + 2 = 10.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
K. Chandrasekharan, Introduction to Analytic Number Theory. Springer-Verlag, 1968, Chap. VI.
K. Chandrasekharan, Arithmetical Functions. Springer-Verlag, 1970, Chapter VIII, pp. 194-228. Springer-Verlag, Berlin.
P. G. L. Dirichlet, Werke, Vol. ii, pp. 49-66.
M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 7.
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
Hari Kishan, Number Theory, Krishna, Educational Publishers, 2014, Theorem 1, p. 133.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 56.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Nurdin N. Takenov and Oksana Haritonova, Representation of positive integers by a special set of digits and sequences, in Dolmatov, S. L. et al. editors, Materials of Science, Practical seminar "Modern Mathematics".
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.6.13 on page 107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 1000 terms from T. D. Noe)
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanța, Vol. 22(1),2014, 13-23.
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
D. Berkane, O. Bordellès, and O. Ramaré, Explicit upper bounds for the remainder term in the divisor problem, Math. Comp. 81:278 (2012), pp. 1025-1051.
Peter J. Cameron and Hamid Reza Dorbidi, Minimal cover groups, arXiv:2311.15652 [math.GR], 2023. See p. 13.
Diophante, A1712, La même parité (in French).
Xiaoxi Duan and M. W. Wong, The Dirichlet divisor problem, traces and determinants for complex powers of the twisted bi-Laplacian, J. of Math. Analysis and Applications, Volume 410, Issue 1, Feb 01 2014, Pages 151-157
L. Hoehn and J. Ridenhour, Summations involving computer-related functions, Math. Mag., 62 (1989), 191-196.
M. N. Huxley, Exponential Sums and Lattice Points III, Proc. London Math. Soc., 87 (2003), pp. 591-609.
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Richard Sladkey, A successive approximation algorithm for computing the divisor summatory function, arXiv:1206.3369 [math.NT], 2012.
Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Math. Comp. 81 (2012), 1233-1246; also at arXiv:1009.3956 [math.NT], 2010-2012.

Right edge of A056535. Cf. A000005, A001659, A052511, A143236.
Row sums of triangle A003988, A010766 and A143724.
A061017 is an inverse.
It appears that the partial sums give A078567. - N. J. A. Sloane, Nov 24 2008
Cf. A116477, A051731, A055507, A161700, A004125, A212120, A341062.

a006218 n = sum $ map (div n) [1..n]
-- Reinhard Zumkeller, Jan 29 2011

[0] cat [&+[Floor(n/k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

with(numtheory): A006218 := n->add(sigma[0](i), i=1..n);

Table[Sum[DivisorSigma[0, k], {k, n}], {n, 70}]
FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]] //Rest (* much faster *)
Join[{0},Accumulate[DivisorSigma[0,Range[60]]]] (* Harvey P. Dale, Jan 06 2016 *)

PARI
```
a(n)=sum(k=1,n,n\k)
```

a(n)=sum(k=1,sqrtint(n),n\k)*2-sqrtint(n)^2 \\ Charles R Greathouse IV, Oct 10 2010

from sympy import integer_nthroot
def A006218(n): return 2*sum(n//k for k in range(1,integer_nthroot(n,2)[0]+1))-integer_nthroot(n,2)[0]**2 # Chai Wah Wu, Mar 29 2021

a(n) = n * ( log(n) + 2*gamma - 1 ) + O(sqrt(n)), where gamma is the Euler-Mascheroni number ~ 0.57721... (see A001620), Dirichlet, 1849. Again, a(n) = n * ( log(n) + 2*gamma - 1 ) + O(log(n)*n^(1/3)). The determination of the precise size of the error term is an unsolved problem (the so-called Dirichlet divisor problem) - see references, especially Huxley (2003).
The bounds from Chandrasekharan lead to the explicit bounds n log(n) + (2 gamma - 1) n - 4 sqrt(n) - 1 <= a(n) <= n log(n) + (2 gamma - 1) n + 4 sqrt(n). - David Applegate, Oct 14 2008
a(n) = 2*(Sum_{i=1..floor(sqrt(n))} floor(n/i)) - floor(sqrt(n))^2. - Benoit Cloitre, May 12 2002
G.f.: (1/(1-x))*Sum_{k >= 1} x^k/(1-x^k). - Benoit Cloitre, Apr 23 2003
For n > 0: A027750(a(n-1) + k) = k-divisor of n, = k <= A000005(n). - Reinhard Zumkeller, May 10 2006
a(n) = A161886(n) - n + 1 = A161886(n-1) - A049820(n) + 2 = A161886(n-1) + A000005(n) - n + 2 = A006590(n) + A000005(n) - n = A006590(n+1) - n - 1 = A006590(n) + A000005(n) - n for n >= 2. a(n) = a(n-1) + A000005(n) for n >= 1. - Jaroslav Krizek, Nov 14 2009
D(n) = Sum_{m >= 2, r >= 1} (r/m^(r+1)) * Sum_{j = 1..m - 1} * Sum_{k = 0 .. m^(r+1) - 1} exp{ 2*k*pi i(p^n + (m - j)m^r) / m^(r+1) } where p is some fixed prime number. - A. Neves, Oct 04 2010
Let E(n) = a(n) - n(log n + 2 gamma - 1). Then Berkane-Bordellès-Ramaré show that |E(n)| <= 0.961 sqrt(n), |E(n)| <= 0.397 sqrt(n) for n > 5559, and |E(n)| <= 0.764 n^(1/3) log n for x > 9994. - Charles R Greathouse IV, Jul 02 2012
a(n) = Sum_{k = 1..floor(sqrt(n))} A005408(floor((n/k) - (k-1))). - Gregory R. Bryant, Apr 20 2013
Dirichlet g.f. for s > 2: Sum_{n>=1} a(n)/n^s = Sum_{k>=1} (Zeta(s-1) - Sum_{n=1..k-1} (HurwitzZeta(s,n/k)*n/k^s))/k. - Mats Granvik, Sep 24 2017
From Ridouane Oudra, Dec 31 2022: (Start)
a(n) = n^2 - Sum_{i=1..n} Sum_{j=1..n} floor(log(i*j)/log(n+1));
a(n) = floor(sqrt(n)) + 2*Sum_{i=1..n} floor((sqrt(i^2 + 4*n) - i)/2);
a(n) = n + Sum_{i=1..n} v_2(i)*round(n/i), where v_2(i) = A007814(i). (End)

A211266 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 12, 15, 18, 21, 24, 28, 30, 34, 38, 41, 44, 49, 51, 56, 60, 63, 67, 72, 75, 79, 83, 88, 91, 97, 99, 104, 109, 112, 117, 123, 125, 130, 135, 140, 143, 149, 152, 157, 163, 167, 170, 177, 180, 186, 190, 194, 199, 205, 209, 215, 219, 223

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

Guide to related sequences:

A056924 ... 1<=x

A211159 ... 1<=x

A211261 ... 1<=x

A211262 ... 1<=x

A211263 ... 1<=x

A211264 ... 1<=x

A211265 ... 1<=x

A211266 ... 1<=x

A211267 ... 1<=x

A181972 ... 1<=x

A038548 ... 1<=x<=y<=n ... x*y=n

A072670 ... 1<=x<=y<=n ... x*y=n+1

A211270 ... 1<=x<=y<=n ... x*y=2n

A211271 ... 1<=x<=y<=n ... x*y=3n

A211272 ... 1<=x<=y<=n ... x*y=floor(n/2)

A094820 ... 1<=x<=y<=n ... x*y<=n

A091627 ... 1<=x<=y<=n ... x*y<=n+1

A211273 ... 1<=x<=y<=n ... x*y<=2n

A211274 ... 1<=x<=y<=n ... x*y<=3n

A211275 ... 1<=x<=y<=n ... x*y<=floor(n/2)

			a(6) counts these pairs: (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,4).

Cf. A211261, A211264.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A094820 Partial sums of A038548.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 13, 15, 16, 19, 20, 22, 24, 27, 28, 31, 32, 35, 37, 39, 40, 44, 46, 48, 50, 53, 54, 58, 59, 62, 64, 66, 68, 73, 74, 76, 78, 82, 83, 87, 88, 91, 94, 96, 97, 102, 104, 107, 109, 112, 113, 117, 119, 123, 125, 127, 128, 134, 135, 137, 140, 144, 146, 150

Offset: 1

Vladeta Jovovic, Jun 12 2004

a(n) = number of pairs (c,d) of integers such that 0 < c <= d and c*d <= n. - Clark Kimberling, Jun 18 2011

Equivalently, the number of representations of n in the form x + y*z, where x, y, and z are positive integers and y <= z. - John W. Layman, Feb 21 2012

Peter Kagey, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A091626, A038548.

Cf. A006218, A000196, A211264.

ListTools:-PartialSums([seq(ceil(numtheory:-tau(n)/2), n=1..100)]); # Robert Israel, Feb 24 2016

f[n_, k_] := Floor[n/k] - Floor[(n - 1)/k]
g[n_, k_] := If[k^2 <= n, f[n, k], 0]
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 100}] (* A000005 *)
t = Table[Sum[g[n, k], {k, 1, n}], {n, 1, 100}]
(* A038548 *)
a[n_] := Sum[t[[i]], {i, 1, n}]
Table[a[n], {n, 1, 100}]  (* A094820 *)
(* Clark Kimberling, Jun 18 2011 *)
Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* Jean-François Alcover, Dec 13 2012 *)

a(n) = sum(k=1, n, ceil(numdiv(k)/2)); \\ Michel Marcus, Feb 24 2016

from math import isqrt
def A094820(n): return ((s:=isqrt(n))*(1-s)>>1)+sum(n//k for k in range(1,s+1)) # Chai Wah Wu, Oct 23 2023

def a(n)
    (1..Math.sqrt(n)).inject(0) { |accum, i| accum + 1 + (n/i).to_i - i }
  end # Peter Kagey, Feb 24 2016

G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^2)/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ (log(n) + 2*gamma - 1)*n/2 + sqrt(n)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 19 2019
a(n) = (A006218(n) + A000196(n))/2. - Ridouane Oudra, Nov 25 2022
a(n) = A211264(n) + A000196(n). - Ridouane Oudra, Sep 13 2024

A211159 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 3, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 2, 1, 3, 0, 2, 1, 3, 0, 5, 0, 1, 2, 2, 1, 3, 0, 4, 1, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 2, 1, 1, 1, 5, 0, 2, 2, 3

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

			a(11) counts these pairs: (2,6), (3,4).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A010052, A200213, A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1}, {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211159(n) = (numdiv(1+n)-issquare(1+n)-2)/2; \\ Antti Karttunen, Jul 07 2017

(define (A211159 n) (/ (- (A000005 (+ 1 n)) (A010052 (+ 1 n)) 2) 2)) ;; Antti Karttunen, Jul 07 2017

a(n) = (A000005(1+n) - A010052(1+n) - 2)/2 = A200213(1+n)/2. - Antti Karttunen, Jul 07 2017

A211261 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A099777, A211266, A211270.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211261(n) = sumdiv(2*n,y,(((2*n/y)Antti Karttunen, Sep 30 2018

a(n) = numdiv(n<<1)>>1-1 \\ David A. Corneth, Sep 30 2018

a(n) = floor(A000005(2*n)/2)-1. - Antti Karttunen, Sep 30 2018, after David A. Corneth's PARI-program

A211262 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A211266

Cf. also A211271.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211262(n) = { my(n3=3*n); sumdiv(n3,d,(d < (n3/d) && (n3/d) <= n)); }; \\ Antti Karttunen, Jan 15 2025

Data section extended up to a(108) by Antti Karttunen, Jan 15 2025

A181972 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 7, 9, 9, 10, 10, 12, 12, 13, 13, 16, 16, 17, 17, 19, 19, 21, 21, 23, 23, 24, 24, 27, 27, 28, 28, 31, 31, 33, 33, 35, 35, 36, 36, 40, 40, 41, 41, 43, 43, 45, 45, 48, 48, 49, 49, 53, 53, 54, 54, 57, 57, 59, 59, 61, 61, 63, 63, 67

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211263 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 3, 3, 1, 1, 4, 4, 1, 1, 3, 3, 2, 2, 2, 2, 2, 2, 4, 4, 1, 1, 2, 2, 2, 2, 4, 4, 1, 1, 4, 4, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 5, 5, 1, 1

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

			a(12) counts these pairs: (1,6) and (2,3).

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211265 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 15, 16, 18, 20, 22, 23, 26, 27, 30, 32, 34, 35, 39, 40, 42, 44, 47, 48, 52, 53, 56, 58, 60, 62, 66, 67, 69, 71, 75, 76, 80, 81, 84, 87, 89, 90, 95, 96, 99, 101, 104, 105, 109, 111, 115, 117, 119, 120, 126, 127, 129, 132, 135, 137

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211267 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 16, 20, 23, 28, 32, 37, 40, 46, 51, 56, 60, 65, 71, 77, 81, 87, 91, 99, 103, 109, 115, 121, 125, 133, 138, 145, 150, 156, 163, 169, 174, 181, 187, 196, 199, 207, 212, 220, 226, 232, 239, 247, 252, 259, 265, 274, 277, 287, 293, 301, 307

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

			a(5) counts these pairs: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211266.

N:= 100: # for a(1)..a(N)
L:= Vector(N):
for x from 1 to floor(sqrt(N)) do
   for y from x+1 while y<=N and x*y<=3*N do
     n0:= max(y, ceil(x*y/3));
     L[n0]:= L[n0]+1;
od od:
ListTools:-PartialSums(convert(L,list)); # Robert Israel, Oct 18 2019

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

Showing 1-10 of 10 results.

cp's OEIS Frontend

A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

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A211266 Number of integer pairs (x,y) such that 0

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A094820 Partial sums of A038548.

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A211159 Number of integer pairs (x,y) such that 0

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A211261 Number of integer pairs (x,y) such that 0

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A211262 Number of integer pairs (x,y) such that 0

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A181972 Number of integer pairs (x,y) such that 0

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