A072670 -id:A072670 - cp's OEIS frontend

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

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

A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 13, 15, 15, 16, 16, 19, 19, 20, 20, 22, 22, 24, 24, 27, 27, 28, 28, 31, 31, 32, 32, 35, 35, 37, 37, 39, 39, 40, 40, 44, 44, 46, 46, 48, 48, 50, 50, 53, 53, 54, 54, 58, 58, 59, 59, 62, 62, 64, 64, 66, 66, 68, 68

Offset: 1

Clark Kimberling, Apr 07 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, 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 *)

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

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

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A260804 Number of ways to write n as n = x * y * z * t + x + y + z + t where 1 <= x <= y <= z <= t <= n.

Original entry on oeis.org

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

Offset: 0

David A. Corneth, Jul 31 2015

nonn

a(n) = A071689(n) - A001399(n) = A071689(n) - round((n+3)^2/12).
From Vladimir Shevelev, Aug 03 2015: (Start)
Is the set of n for which a(n)=0 finite?
Note that this set contains only numbers n of the form prime + 1. Indeed, if n-1>=4 is a composite number, then n = p*q + 1, p>=2, q>=2. If p <= q, then, for x=1, y=1, z = p-1, t = q-1, we have
x*y*z*t + x + y + z + t = 1*1*(p-1)*(q-1) + 1 + 1 + (p-1) + (q-1) = p*q + 1 = n; so a(n) >= 1. If p > q, then we set x=1, y=1, z = q-1, t = p-1, and again a(n) >= 1.
Note also that limsup_{n->infinity} (a(n)) = infinity. Indeed, this limit is realized, say, on n = primorials +1 (A002110), since, when m goes to infinity, the number of representations of n - 1 = A002110(m) of the form p*q tends to infinity. On primorials +1 > 2 we have a subsequence: 0,1,3,8,27,... .
A generalization. For k>=2, let b_k(n) be the number of ways to write n as n = x_1 * x_2 *...* x_k + x_1 + x_2 + ... + x_k, where 1 <= x_1 <= x_2 <= ... <= x_k <= n.
Then, for n >= k-1, b_k(n) = 0 yields that n - k + 3 is prime with similar other comments. In particular, only b_2(n) = 0 if and only if n+1 is 1 or prime (cf. A072670). (End)

David A. Corneth, Table of n, a(n) for n = 0..9999
Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Cf. A001399, A071689, A260803, A260965.

xmax = 9; ymax = 21; zmax = 98; (* When extending data, terms where maxima for x, y or z are reached have to be checked one by one. *)
r[n_] := r[n] = Module[{r1, r2, r3, rn}, r1 = Reap[Do[rn = Reduce[n == x y z t + x + y + z + t && 1 <= x <= y <= z <= t <= n, t, Integers]; If[rn =!= False, Sow[{x, y, z, t} /. {ToRules[rn]}]], {x, 1, xmax}, {y, 1, ymax}, {z, 1, zmax}]]; If[r1 == {Null, {}} , {}, r2 = r1[[2, 1]]; r3 = Flatten[r2, 1]; If[Max[r3[[All, 1]]] == xmax, Print[ "xmax reached at n = ", n]]; If[Max[r3[[All, 2]]] == ymax, Print["ymax reached at n = ", n]]; If[Max[r3[[All, 3]]] == zmax, Print["zmax reached at n = ", n]]; r3]];
a[n_] := Length[r[n]];
Table[Print["a(", n, ") = ", a[n], " ", r[n]]; a[n], {n, 0, 109}] (* Jean-François Alcover, Nov 19 2018 *)

If A260803(n) > 0, then a(n+1) > 0. So if a(n+1) = 0, then A260803(n) = 0. Converse statement is not true. For example, a(24) > 0, while A260803(23) = 0. - Vladimir Shevelev, Aug 14 2015

A067432 Number of ways to represent the n-th prime in form p*q+p+q, where p and q are primes (see A066938).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 15 2002

nonn

a(A049084(A066938(n))) > 0; a(A049084(A198273(n))) = 0; a(A049084(A198277(n))) = n and a(A049084(m)) <> n for m < A198277(n). [Reinhard Zumkeller, Oct 23 2011]

a(n) < A072670(n).

			a(15) = 2 as A000040(15) = 47 = 3*11+3+11 = 5*7+5+7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000196, A010051.

a067432 n = length [p | let prime_n = a000040 n,
   p <- takeWhile (< a000196 prime_n) a000040_list,
   let (q,m) = divMod (prime_n - p) (p + 1),
   m == 0, a010051 q == 1]
a067432_list = map a067432 [1..]
-- Reinhard Zumkeller, Oct 23 2011

A081757 Number of ways to write n as i*j+i-j, 0

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

			n=25 = 2*23+2-23 = 3*11+3-11 = 4*7+4-7, therefore a(25)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A072670.

a081757 n = length [() | j <- [2..n], i <- [1..j-1], i * j + i - j == n]
-- Reinhard Zumkeller, May 24 2013

G.f.: Sum(x^(n^2+3*n)/(1-x^n),n = 1 .. infinity). - Vladeta Jovovic, May 13 2008

cp's OEIS Frontend

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

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

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

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A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

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A347708 Number of distinct possible alternating products of odd-length factorizations of n.

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