A067721 -id:A067721 - cp's OEIS frontend

A159899 A variant of A067721 with a different beginning; see A067721 for further information.

1, 4, 2, 9, 1, 3, 8, 25, 4, 36, 18, 1, 2, 64, 6, 81, 16, 4, 50, 121, 1, 20, 72, 9, 36, 196, 2, 225, 4, 11, 128, 1, 12, 324, 162, 13, 5, 400, 8, 441, 100, 3, 242, 529, 1, 63, 40, 17, 144, 676, 18, 9, 7, 19, 392, 841, 4, 900, 450, 12, 8, 16, 22, 1089, 256, 23, 2, 1225

Offset: 1

Ctibor O. Zizka, Apr 25 2009

Cf. A067721.

A115878 a(n) is the number of positive solutions of the Diophantine equation x^2 = y(y+n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 4, 1, 1, 4, 2, 1, 3, 1, 1, 4, 1, 3, 4, 1, 4, 2, 1, 1, 4, 4, 1, 4, 1, 1, 7, 1, 1, 7, 2, 2, 4, 1, 1, 3, 4, 4, 4, 1, 1, 4, 1, 1, 7, 4, 4, 4, 1, 1, 4, 4, 1, 7, 1, 1, 7, 1, 4, 4, 1, 7, 4, 1, 1, 4, 4, 1, 4, 4, 1, 7, 4, 1, 4, 1, 4, 10, 1, 2, 7, 2, 1, 4

Offset: 1

Giovanni Resta, Feb 02 2006

Number of divisors d of n^2 such that d^2 < n^2 and n^2/d == d (mod 4). - Antti Karttunen, Oct 06 2018, based on Robert Israel's Jun 27 2014 comment in A115880.
For odd n, a(n) can be computed from the prime signature. - David A. Corneth, Oct 07 2018
Number of r X s rectangles with integer side lengths such that r + s = n, r < s and (s-r) | (s*r). - Wesley Ivan Hurt, Apr 24 2020

			a(15) = 4 since there are 4 solutions (x,y) to x^2 = y(y+15), namely (4,1), (10,5), (18, 12) and (56, 49).
Note how each x is obtained from each such divisor pair n2/d and d of n2 as (n2/d - d)/4, when their difference is a positive multiple of four, thus in case of n2 = 15^2 = 225 we get (225/1 - 1)/4 = 56, (225/3 - 3)/4 = 18, (225/5 - 5) = 10 and (225/9 - 9)/4 = 4. - _Antti Karttunen_, Oct 06 2018
a(96) = 10. We compute P, the largest power of 2 dividing n = 96. Then compute min(P, 4) and divide n by it. This gives 96/4 = 24. Then find the number of divisors of 24^2, which is 21. Dividing by 2 rounding down to the nearest integer gives 10, the value of a(96). - _David A. Corneth_, Oct 06 2018

Antti Karttunen, Table of n, a(n) for n = 1..18480
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A067721, A115879, A115880, A115881.

a[n_] := Sum[Boole[d^2 < n^2 && Mod[n^2/d-d, 4] == 0], {d, Divisors[n^2]}];
Array[a, 102] (* Jean-François Alcover, Feb 27 2019, from PARI *)

A115878(n) = { my(n2 = n^2); sumdiv(n2,d,((d*d)Antti Karttunen, Oct 06 2018

a(n) = my(v=min(2, valuation(n,2))); numdiv((n>>v)^2)>>1 \\ David A. Corneth, Oct 06 2018

from itertools import takewhile
from sympy import divisors
def A115878(n): return sum(1 for d in takewhile(lambda d:dChai Wah Wu, Aug 21 2024

From David A. Corneth, Oct 07 2018: (Start)
a((2k+1) * 2^m) = floor(tau((2k + 1) ^ 2) / 2) for m <= 2.
a((2k+1) * 2^m) = (2m - 3) * a(2k+1) + (m-2) for m > 2. (End)
a(n) = Sum_{i=1..floor((n-1)/2)} (1 - ceiling(i*(n-i)/(n-2*i)) + floor(i*(n-i)/(n-2*i))). - Wesley Ivan Hurt, Apr 24 2020

A115879 a(n) is the least positive x satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 12, 3, 6, 12, 30, 8, 42, 24, 4, 6, 72, 12, 90, 24, 10, 60, 132, 5, 30, 84, 18, 48, 210, 8, 240, 12, 22, 144, 6, 24, 342, 180, 26, 15, 420, 20, 462, 120, 12, 264, 552, 7, 84, 60, 34, 168, 702, 36, 24, 21, 38, 420, 870, 16, 930, 480, 8, 24, 36, 44

Offset: 1

Giovanni Resta, Feb 02 2006

			a(15)=4 since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The least x values is 4, from (x,y)=(4,1).

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

Cf. A067721, A115878, A115880, A115881.

f:= proc(n) local q;
  q:= max(select(t -> t^2 < n^2 and n^2/t - t mod 4 = 0, numtheory:-divisors(n^2)));
  if q = -infinity then 0 else (n^2/q - q)/4 fi;
end proc:
map(f, [$1..100]); # Robert Israel, Aug 21 2025

from itertools import takewhile
from collections import deque
from sympy import divisors
def A115879(n): return -(a:=next(iter(deque((d for d in takewhile(lambda d:d>2 # Chai Wah Wu, Aug 21 2024

A115880 Largest positive x satisfying the Diophantine equation x^2 = y*(y+n), a(n)=0 if there are no solutions.

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 12, 3, 20, 12, 30, 8, 42, 24, 56, 15, 72, 40, 90, 24, 110, 60, 132, 35, 156, 84, 182, 48, 210, 112, 240, 63, 272, 144, 306, 80, 342, 180, 380, 99, 420, 220, 462, 120, 506, 264, 552, 143, 600, 312, 650, 168, 702, 364, 756, 195, 812, 420, 870

Offset: 1

Giovanni Resta, Feb 02 2006

nonn

Notice that x^2 = y*(y+n) is equivalent to (n+2*y+2*x)*(n+2*y-2*x) = n^2. We take the factorization of n^2 into two factors congruent mod 4 where one is as small as possible and the other is as large as possible. For n == 0 mod 4 the factors are 4 and n^2/4, for n == 2 mod 4 they are 2 and n^2/2, for n odd they are 1 and n^2. - Robert Israel, Jun 27 2014

			a(15)=56 since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The largest x is 56, from (x,y)=(56,49).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A067721, A115878, A115879, A115881.

Table[Max[x/.Solve[{x^2==y(y+n),x>0},{x,y},Integers]],{n,1,100}]/.x->0 (* Vaclav Kotesovec, Jun 26 2014 *)

def A115880(n):
    a, b = divmod(n,4)
    return (a**2-1,(c:=a<<1)*(c+1),c*(a+1),c*(c+3)+2)[b] # Chai Wah Wu, Aug 21 2024

Empirical g.f.: x^3*(x^9-2*x^6-3*x^5-6*x^4-4*x^3-6*x^2-2) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014
a(4*j) = j^2 - 1,
a(4*j+1) = 4*j^2+2*j,
a(4*j+2) = 2*j^2+2*j,
a(4*j+3) = 4*j^2+6*j+2. (see Comments) - Robert Israel, Jun 27 2014

A115881 a(n) is the largest positive y satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.

Original entry on oeis.org

0, 0, 1, 0, 4, 2, 9, 1, 16, 8, 25, 4, 36, 18, 49, 9, 64, 32, 81, 16, 100, 50, 121, 25, 144, 72, 169, 36, 196, 98, 225, 49, 256, 128, 289, 64, 324, 162, 361, 81, 400, 200, 441, 100, 484, 242, 529, 121, 576, 288, 625, 144, 676, 338, 729, 169, 784, 392, 841, 196

Offset: 1

Giovanni Resta, Feb 02 2006

nonn

The corresponding least y is given by A067721(n).

			a(15)=49, since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The largest y is 49, from (x,y)=(56,49).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A067721, A115878, A115879, A115880.

Table[Max[y/.Solve[{x^2==y*(y+n),y>0},{x,y},Integers]],{n,1,100}]/.y->0 (* Vaclav Kotesovec, Jun 26 2014 *)

def A115881(n):
    a, b = divmod(n,4)
    return ((c:=a**2)-(a<<1)+1,(d:=c<<2),c<<1,d+(a<<2)+1)[b] # Chai Wah Wu, Aug 21 2024

Empirical g.f.: -x^3*(x^9+x^8+2*x^7+4*x^6+x^5+6*x^4+2*x^3+4*x^2+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014
From empirical g.f.: a(n) = 1/2 - n/2 + 11*n^2/64 + (1/4 - 1/32*n^2) * (2*floor(n/4) + 2*floor((n+1)/4) - n + 1) + (1/4 - 5/64*n^2)*(-1)^n. - Vaclav Kotesovec, Jun 26 2014
From Chai Wah Wu, Aug 21 2024: (Start)
a(4*j)   = j^2 - 2*j + 1,
a(4*j+1) = 4*j^2,
a(4*j+2) = 2*j^2,
a(4*j+3) = 4*j^2+4*j+1 (see A115880).
(End)

A173636 Number of positive solutions of equation x(x+n)=y*y.

Original entry on oeis.org

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

Offset: 0

Carmine Suriano, Nov 23 2010

nonn

The solution x=y=0 is not counted.

Same as A115878 except for a(0). - Georg Fischer, Oct 12 2018

			a(9)=2 since x(x+9)=y*y has 2 solutions: x=3, y=6 and x=16, y=20.

Cf. A067721, A115878.

Join[{0}, Table[Length[{ToRules[Reduce[x (x + n) == y^2 && x > 0 && y > 0, {x, y}, Integers]]}], {n, 100}]]

a(n) = A176835(n)-1. - R. J. Mathar, Nov 23 2010

cp's OEIS Frontend

A159899 A variant of A067721 with a different beginning; see A067721 for further information.

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A115878 a(n) is the number of positive solutions of the Diophantine equation x^2 = y(y+n).

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A115879 a(n) is the least positive x satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.

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A115880 Largest positive x satisfying the Diophantine equation x^2 = y*(y+n), a(n)=0 if there are no solutions.

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A115881 a(n) is the largest positive y satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.

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Keywords

Comments

Examples

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Crossrefs

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Mathematica

Python

Formula

A173636 Number of positive solutions of equation x(x+n)=y*y.

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Comments

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Mathematica

Formula