A211261 -id:A211261 - cp's OEIS frontend

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 *)

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

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)

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 *)

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

Showing 1-9 of 9 results.

cp's OEIS Frontend

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

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

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

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

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

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