A094820 -id:A094820 - cp's OEIS frontend

A038548 Number of divisors of n that are at most sqrt(n).

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 4, 1, 4, 4

Offset: 1

Tom Verhoeff, N. J. A. Sloane

Number of ways to arrange n identical objects in a rectangle, modulo rotation.
Number of unordered solutions of x*y = n. - Colin Mallows, Jan 26 2002
Number of ways to write n-1 as n-1 = x*y + x + y, 0 <= x <= y <= n. - Benoit Cloitre, Jun 23 2002
Also number of values for x where x+2n and x-2n are both squares (e.g., if n=9, then 18+18 and 18-18 are both squares, as are 82+18 and 82-18 so a(9)=2); this is because a(n) is the number of solutions to n=k(k+r) in which case if x=r^2+2n then x+2n=(r+2k)^2 and x-2n=r^2 (cf. A061408). - Henry Bottomley, May 03 2001
Also number of sums of sequences of consecutive odd numbers or consecutive even numbers including sequences of length 1 (e.g., 12 = 5+7 or 2+4+6 or 12 so a(12)=3). - Naohiro Nomoto, Feb 26 2002
Number of partitions whose consecutive parts differ by exactly two.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 06 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly twice. Example: a(12)=3 because we have [3,3,2,2,1,1],[2,2,2,2,2,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
a(n) is also the number of nonnegative integer solutions of the Diophantine equation 4*x^2 - y^2 = 16*n. For example, a(24)=4 because there are 4 solutions: (x,y) = (10,4), (11,10), (14,20), (25,46). - N-E. Fahssi, Feb 27 2008
a(n) is the number of even divisors of 2*n that are <= sqrt(2*n). - Joerg Arndt, Mar 04 2010
First differences of A094820. - John W. Layman, Feb 21 2012
a(n) = #{k: A027750(n,k) <= A000196(n)}; a(A008578(n)) = 1; a(A002808(n)) > 1. - Reinhard Zumkeller, Dec 26 2012
Row lengths of the tables in A161906 and A161908. - Reinhard Zumkeller, Mar 08 2013
Number of positive integers in the sequence defined by x_0 = n, x_(k+1) = (k+1)*(x_k-2)/(k+2) or equivalently by x_k = n/(k+1) - k. - Luc Rousseau, Mar 03 2018
Expanding the first comment: Number of rectangles with area n and integer side lengths, modulo rotation. Also number of 2D grids of n congruent squares, in a rectangle, modulo rotation (cf. A000005 for rectangles instead of squares; cf. A034836 for the 3D case). - Manfred Boergens, Jun 08 2021
Number of divisors of n that have an even number of prime divisors (counted with multiplicity), or in other words, number of terms of A028260 that divide n. - Antti Karttunen, Apr 17 2022

			a(4) = 2 since 4 = 2 * 2 = 4 * 1. Also A034178(4*4) = 2 since 16 = 4^2 - 0^2 = 5^2 - 3^2. - _Michael Somos_, May 11 2011
x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + ...

George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, page 18, exer. 21, 22.

T. D. Noe, Table of n, a(n) for n = 1..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 17-34.
Christopher Briggs, Y. Hirano, and H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.4.
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, (2.27).
Madeline Locus Dawsey, Matthew Just and Robert Schneider, A "supernormal" partition statistic, arXiv:2107.14284 [math.NT], 2021. See Table 2 p. 21.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Index entries for sequences computed from exponents in factorization of n.

Different from A068108. Records give A038549, A004778, A086921.
Cf. A000005, A001620, A028260, A056924, A072670, A094820, A161841, A108504.
Cf. A066839, A033676, row sums of A303300.
Inverse Möbius transform of A065043.
Cf. A244664 (Dgf at s=2), A244665 (Dgf at s=3).

a038548 n = length $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Dec 26 2012
(C#)
int A038548(int n)  {
    System.Numerics.BigInteger erg = 0, i;
    for (i = 1; i * i <= n; i++)
        if (n % i == 0) erg++;
    return (int)erg;
} // Frank Hollstein, Jan 08 2023

with(numtheory): A038548 := n->ceil(sigma[0](n)/2);

Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] (* Robert G. Wilson v, Mar 02 2009 *)
Table[Count[Divisors[n],?(#<=Sqrt[n]&)],{n,110}] (* _Harvey P. Dale, Jul 10 2021 *)
Table[Sum[If[n > k*(k-1), 1, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))} /* Michael Somos, Jan 25 2005 */

a(n)=ceil(numdiv(n)/2) \\ Charles R Greathouse IV, Sep 28 2012

from sympy import divisor_count
def A038548(n): return divisor_count(n)+1>>1 # Chai Wah Wu, Dec 19 2023

a(n) = ceiling(d(n)/2), where d(n) = number of divisors of n (A000005).
a(2k) = A034178(2k) + A001227(k). a(2k+1) = A034178(2k+1). - Naohiro Nomoto, Feb 26 2002
G.f.: Sum_{k>=1} x^(k^2)/(1-x^k). - Jon Perry, Sep 10 2004
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/2. - Christian G. Bower, Jun 06 2005 [corrected by Vaclav Kotesovec, Aug 19 2019]
a(n) = (A000005(n) + A010052(n))/2. - Omar E. Pol, Jun 23 2009
a(n) = A034178(4*n). - Michael Somos, May 11 2011
2*a(n) = A161841(n). - R. J. Mathar, Mar 07 2021
a(n) = A000005(n) - A056924(n) = A056924(n) + A010052(n) = Sum_{d|n} A065043(d). - Antti Karttunen, Apr 17 2022
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

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 *)

A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 8, 10, 12, 14, 15, 18, 19, 21, 23, 26, 27, 30, 31, 34, 36, 38, 39, 43, 45, 47, 49, 52, 53, 57, 58, 61, 63, 65, 67, 72, 73, 75, 77, 81, 82, 86, 87, 90, 93, 95, 96, 101, 103, 106, 108, 111, 112, 116, 118, 122, 124, 126, 127, 133, 134, 136, 139, 143

Offset: 0

Eric W. Weisstein, Jan 24 2004

nonn

Also number of ordered pairs of positive integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A000005, A067274, A091626, A094820.

Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]]  (* Jean-François Alcover, Oct 23 2012, after Vladeta Jovovic *)

a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021

N=66; x='x+O('x^N); concat([0, 0], Vec((-x+sum(k=1, sqrtint(N), x^k^2/(1-x^k)))/(1-x))) \\ Seiichi Manyama, Sep 04 2021

from math import isqrt
def A091627(n):
    m = isqrt(n)
    return 0 if n == 0 else sum(n//k for k in range(1, m+1))-m*(m-1)//2-1 # Chai Wah Wu, Oct 07 2021

a(n) = A091626(n) - n - 1. a(n) = a(n-1) + ceiling(tau(n)/2), n>1. Partial sums of A038548. - Vladeta Jovovic, Jun 12 2004

G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - Seiichi Manyama, Sep 04 2021

A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 16, 17, 19, 21, 23, 24, 27, 28, 31, 33, 35, 36, 40, 41, 43, 45, 48, 49, 53, 54, 57, 59, 61, 63, 67, 68, 70, 72, 76, 77, 81, 82, 85, 88, 90, 91, 96, 97, 100, 102, 105, 106, 110, 112, 116, 118, 120, 121, 127, 128, 130, 133, 136

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

Partial sums of A056924.

For a guide to related sequences, see A211266.

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

Cf. A211266, A001222, A006218, A000196.

Cf. A066829, A094820, A181972.

[0] cat [&+[(&+[p[2]: p in Factorization(i)] mod 2) *Floor(n div i):i in [2..n] ]:n in [2..65]]; // Marius A. Burtea, Oct 17 2019

with(numtheory): seq(add((bigomega(i) mod 2)*floor(n/i), i=1..n), n=1..60); # Ridouane Oudra, Oct 17 2019
# Alternative:
ListTools:-PartialSums(map(t-> floor(numtheory:-tau(t)/2), [$1..100])); # 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 *)

from math import isqrt
def A211264(n): return (lambda m: sum(n//k for k in range(1, m+1))-m*(m+1)//2)(isqrt(n)) # Chai Wah Wu, Oct 08 2021

a(n) = (1/2)*Sum_{i=1..n} (1 - A008836(i))*floor(n/i). - Enrique Pérez Herrero, Jul 10 2012 [Corrected by Ridouane Oudra, Oct 17 2019]
From Ridouane Oudra, Oct 17 2019: (Start)
a(n) = Sum_{i=1..n} A066829(i)*floor(n/i)
a(n) = (1/2)*(A006218(n) - A000196(n)). (End)
From Ridouane Oudra, Sep 28 2024: (Start)
a(n) = Sum_{k=1..n} floor((sqrt(k^2 + 4*n) - k)/2) ;
a(n) = A094820(n) - A000196(n) ;
a(n) = A181972(2*n). (End)

A211270 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y = 2n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(12) counts these pairs: (2,12), (3,8), (4,6).
For n = 2, only the pair (2,2) satisfies the condition, thus a(2) = 1. - _Antti Karttunen_, Sep 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Theorem 4.2 p. 7.

Cf. A000005, A211266, A211261.

seq(floor((numtheory:-tau(2*n)-1)/2),n=1..100); # Robert Israel, Feb 25 2019

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* this sequence *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

A211270(n) = sumdiv(2*n,y,(((2*n/y)<=y)&&(y<=n))); \\ Antti Karttunen, Sep 30 2018

a(n) = floor((A000005(2n)-1)/2). - Robert Israel, Feb 25 2019

Term a(2) corrected by Antti Karttunen, Sep 30 2018

A211271 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 2, 2, 1, 4, 2, 2, 2, 4, 1, 4, 1, 4, 2, 2, 3, 4, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 6, 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, 4, 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, 6

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(3) counts this pair: (3,3). - _Antti Karttunen_, Jan 15 2025
a(20) counts these pairs: (3,20), (4,15), (5,12), (6,10).

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

Cf. A211266.

Cf. also A211262.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

A211271(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) and a(3) corrected from 0 to 1 by Antti Karttunen, Jan 15 2025

A211272 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(24) counts these pairs: (1,12), (2,6), (3,4).

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

Cf. A000005, A211266.

[0] cat [Ceiling(#Divisors( Floor(n/2))/2):n in [2..100]]; // Marius A. Burtea, Feb 07 2020

[seq(ceil(numtheory:-tau(floor(n/2))/2),n=1..100)]; - Robert Israel, Feb 07 2020

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(n) = ceiling(A000005(floor(n/2))/2). - Robert Israel, Feb 07 2020

A211273 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 15, 19, 22, 25, 28, 32, 35, 39, 43, 46, 49, 55, 57, 62, 66, 69, 73, 78, 82, 86, 90, 95, 98, 104, 106, 112, 117, 120, 125, 131, 133, 138, 143, 148, 152, 158, 161, 166, 172, 176, 179, 186, 189, 196, 200, 204, 209, 215, 219, 225, 229, 233

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

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

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(1)-a(2) corrected by Sean A. Irvine, Jan 22 2025

A211274 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= 3n.

Original entry on oeis.org

1, 3, 6, 9, 12, 16, 20, 24, 28, 33, 37, 43, 46, 52, 57, 62, 67, 72, 78, 84, 88, 95, 99, 107, 111, 117, 124, 130, 134, 142, 147, 154, 159, 166, 173, 179, 184, 191, 197, 206, 210, 218, 223, 231, 237, 243, 250, 259, 264, 271, 277, 286, 289, 299, 305, 313

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

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

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(1)-a(3) corrected by Sean A. Irvine, Jan 22 2025

Showing 1-10 of 16 results. Next

cp's OEIS Frontend

A176708 Partial sums of A094820.

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A038548 Number of divisors of n that are at most sqrt(n).

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

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A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

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A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.

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Magma

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A211270 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y = 2n.

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A211271 Number of integer pairs (x,y) such that 0

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