A055527 -id:A055527 - cp's OEIS frontend

A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

0, 1, 3, 6, 10, 3, 21, 8, 36, 15, 55, 6, 78, 35, 15, 48, 136, 27, 171, 10, 42, 99, 253, 10, 300, 143, 81, 42, 406, 15, 465, 64, 88, 255, 35, 63, 666, 323, 91, 3, 820, 21, 903, 55, 66, 483, 1081, 48, 1176, 125, 85, 39, 1378, 81, 165, 28, 76, 783, 1711, 15, 1830, 899, 63

Offset: 1

Alex Ratushnyak, Nov 19 2013

Numbers n such that a(n) = n*(n-1)/2 appear to be A000430.

n = 1 is the only number for which a(n) = 0. - T. D. Noe, Nov 21 2013

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe).
StackExchange, The cube of integer can be written as the difference of two square.

Cf. A000290, A000430, A000578, A038202, A055527, A232176.

Join[{0}, Table[k = 1; While[! IntegerQ[Sqrt[n^3 + k^2]], k++]; k, {n, 2, 100}]] (* T. D. Noe, Nov 21 2013 *)

a(n) = {k = 1; while (!issquare(n^3+k^2), k++); k;} \\ Michel Marcus, Nov 20 2013

import math
for n in range(77):
   n3 = n*n*n
   y=1
   for k in range(1, 10000001):
     s = n3 + k*k
     r = int(math.sqrt(s))
     if r*r == s:
       print(k, end=', ')
       y=0
       break
   if y: print(end='-, ')

from _future_ import division
from sympy import divisors
def A232175(n):
    n3 = n**3
    ds = divisors(n3)
    for i in range(len(ds)//2-1,-1,-1):
        x = ds[i]
        y = n3//x
        a, b = divmod(y-x,2)
        if not b:
            return a
    return 0 # Chai Wah Wu, Sep 12 2017

A300728 a(n) is the smallest positive number k such that k^2 + k*n + n^2 is a perfect square, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 0, 5, 0, 3, 10, 8, 7, 15, 6, 24, 20, 35, 16, 9, 5, 63, 30, 80, 12, 24, 48, 120, 11, 15, 70, 45, 32, 195, 18, 224, 10, 7, 126, 13, 60, 323, 160, 16, 24, 399, 48, 440, 96, 27, 240, 528, 15, 56, 30, 40, 140, 675, 90, 33, 9, 55, 390, 840, 36, 899, 448, 17, 20, 39, 14, 1088, 252, 91, 26, 1224, 33, 1295, 646, 45

Offset: 0

Altug Alkan, Mar 11 2018

A positive a(n) cannot be 1 or a multiple of n for n > 0 since there is no square in A002061 except 1. Also it is easy to show that a(n) cannot be 2 or 4 since a(2) = a(4) = 0.
From Robert Israel, Mar 12 2018: (Start)
If n >= 5 is odd, a(n) <= n^2/4 - n/2 - 3/4, with a(n) = n^2/4 - n/2 - 3/4 if n is a prime >= 5.
If n >= 10 and n == 2 (mod 4), a(n) <= n^2/8 - n/2 - 3/2, with equality if n/2 is a prime >= 5.
If n >= 16 and n == 0 (mod 4), 1 < a(n) <= n^2/16 - n/2 - 3, with equality if n/4 is 4 or a prime >= 5. (End)

			a(2) = 0 because k^2 + 2*k + 4 = (k + 1)^2 + 3 cannot be a square for k > 0.
a(4) = 0 because k^2 + 4*k + 16 = (k + 2)^2 + 12 cannot be a square for k > 0.
a(5) = 3 because 3^2 + 3*5 + 5^2 = 7^2 and 3 is the least positive number with this property.

Altug Alkan, Table of n, a(n) for n = 0..10000
Altug Alkan, Scatterplot of first differences for n <= 10^4

Cf. A000290, A003136, A055527.

f:= proc(n) local k;
for k from 1 do if issqr(k^2 + k*n + n^2) then return k fi od
end proc:
f(1):= 0: f(2):= 0: f(4):= 0:
map(f, [$0..200]); # Robert Israel, Mar 12 2018

f[n_] := Module[{k},
For[k = 1, True, k++, If[IntegerQ[Sqrt[k^2 + k*n + n^2]], Return[k]]]];
f[1] = 0; f[2] = 0; f[4] = 0;
Map[f, Range[0, 200]] (* Jean-François Alcover, Nov 11 2023, after Robert Israel *)

from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A300728(n): return min((d[0] for d in diop_quadratic(x*(x+n)+n**2-y**2) if d[0]>0), default=0) if n else 1 # Chai Wah Wu, Nov 11 2023

A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 3, 3, 1, 3, 1, 3, 4, 1, 3, 5, 1, 1, 4, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 4, 3, 1, 3, 3, 5, 4, 1, 1, 9, 1, 1, 5, 4, 4, 4, 1, 3, 4, 3, 1

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009004, A055525, A055527.

a(n)=my(b);sum(c=n+2,n^2\2+1,issquare(c^2-n^2,&b) && nCharles R Greathouse IV, Jul 07 2013

a(n) = A046079(n) - A056137(n) = A046081(n) - A046080(n) - A056137(n).

A238376 Smallest other side of an Heronian triangle with a side of length n.

Original entry on oeis.org

4, 3, 3, 5, 15, 5, 10, 6, 13, 5, 4, 13, 4, 10, 8, 15, 20, 7, 10, 26, 123, 7, 3, 3, 29, 15, 5, 5, 68, 20, 25, 15, 8, 15, 12, 25, 10, 9, 9, 20, 61, 15, 12, 75, 250, 14, 105, 6, 4, 5, 4, 45, 12, 25, 60, 10, 68, 10, 11, 39, 16, 40, 7, 35, 85, 7, 29, 9, 447, 21, 9

Offset: 3

Giovanni Resta, Feb 25 2014

nonn

			a(8) = 5 because (8,5,5) is Heronian triangle, and there are no such triangles with one side equal to 8 and another equal to 1, 2, 3 or 4.

Giovanni Resta, Table of n, a(n) for n = 3..1000

Cf. A055527.

area[a_, b_, c_] := Block[{p=(a+b+c)/2}, Sqrt[p*(p-a)*(p-b)*(p-c)]]; a[n_] := Catch@ Block[{b = 0, c}, While[True, b++; Do[If[ IntegerQ[area[n, b, c]], Throw@b], {c, Max[Abs[n-b] + 1, b], n+b-1}]]] /; n > 2; a/@ Range[3,50]

A239975 Least k>0 such that n^2 + (n+k)^2 is a square, or -1 if no such k exists.

Original entry on oeis.org

7, 2, 17, 7, 3, 14, 49, 4, 71, 34, 5, 14, 127, 6, 161, 1, 7, 98, 241, 8, 35, 142, 9, 17, 391, 10, 449, 28, 11, 254, 49, 12, 647, 322, 13, 2, 799, 14, 881, 73, 15, 482, 1057, 7, 119, 70, 17, 113, 1351, 18, 77, 34, 19, 782, 1681, 3, 1799, 898, 21, 56, 7, 22, 2177, 217, 23

Offset: 5

Alex Ratushnyak, Mar 30 2014

nonn

a(3*n) <= n because with k=n: (3*n)^2 + (4*n)^2 = (5*n)^2.

			a(6) = 2 because 6^2 + (6+2)^2 = 100 is a square.
a(20) = 1 because 20^2 + 21^2 = 841 = 29^2.

Michael S. Branicky, Table of n, a(n) for n = 5..10004

Cf. A000290, A055527 (least k>0 such that n^2 + k^2 is a square).

s=[]; for(n=5, 100, k=1; while(!issquare(n^2+(n+k)^2), k++); s=concat(s, k)); s \\ Colin Barker, Mar 31 2014

from sympy.ntheory.primetest import is_square
def a(n):
    k = 1
    while not is_square(n**2 + (n+k)**2): k += 1
    return k
print([a(n) for n in range(5, 70)]) # Michael S. Branicky, Jul 02 2021

A370574 a(n) is the least k > 0 such that n^2 XOR k^2 is a positive square number (where XOR denotes the bitwise XOR operator), or -1 if no such k exists.

Original entry on oeis.org

-1, -1, 4, 3, 3, 8, 24, 6, 23, 6, 44, 16, 43, 48, 36, 12, 111, 46, 180, 12, 72, 88, 9, 7, 7, 86, 20, 59, 20, 72, 56, 24, 479, 222, 20, 15, 20, 360, 15, 24, 183, 144, 13, 11, 11, 18, 943, 14, 1103, 14, 747, 172, 405, 40, 159, 31, 492, 40, 28, 144, 28, 112, 31, 48, 660

Offset: 1

Rémy Sigrist, Feb 22 2024

This sequence has similarities with A055527; here we look for triples (n, k, m) of positive integers such that n^2 XOR k^2 = m^2, there for triples such that n^2 + k^2 = m^2.

Conjecture: a(n) > 0 for any n > 2.

			For n = 6: we have:
  k   6^2 XOR k^2  Positive square?
  --  -----------  ----------------
   1           37  No
   2           32  No
   3           45  No
   4           52  No
   5           61  No
   6            0  No
   7           21  No
   8          100  Yes
So a(6) = 8.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Interactive scatterplot of (x, y, z) such that x^2 XOR y^2 = z^2 and x, y, z < 2^13 [provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes]
Rémy Sigrist, Logarithmic scatterplot of the first 140000 terms

Cf. A055527.

f:= proc(n) local k;
  for k from 1 by 1 do if k <> n and issqr(MmaTranslator[Mma]:-BitXor(n^2, k^2)) then return k fi od
end proc: f(1):= -1: f(2):= -1:
map(f, [$1..100]); # Robert Israel, Mar 01 2024

A370574[n_] := If[n <= 2, -1, Block[{k = 0}, While[++k == n || !IntegerQ[ Sqrt[BitXor[n^2, k^2]]]]; k]]; Array[A370574, 100] (* Paolo Xausa, Mar 01 2024 *)

a(n) = { if (n <= 2, return (-1), for (k = 1, oo, if (k!=n &&  issquare( bitxor(n^2, k^2)), return (k)););); }

from sympy import integer_nthroot
def A370574(n):
    if n <= 2: return -1
    k, n_2 = 1, n**2
    while True:
        if (integer_nthroot(n_2 ^ k**2, 2))[1] and k != n: return k
        k += 1 # Karl-Heinz Hofmann, Mar 03 2024

A066662 Shortest leg of a Pythagorean triangle with a hypotenuse of length 5n.

Original entry on oeis.org

3, 6, 9, 12, 7, 18, 21, 24, 27, 14, 33, 36, 16, 42, 21, 48, 13, 54, 57, 28, 63, 66, 69, 72, 35, 32, 81, 84, 17, 42, 93, 96, 99, 26, 49, 108, 57, 114, 48, 56, 45, 126, 129, 132, 63, 138, 141, 144, 147, 70, 39, 64, 23, 162, 77, 168, 171, 34, 177, 84, 55, 186, 189, 192

Offset: 1

Henry Bottomley, Jan 14 2002

nonn

a(n) has values in the range from sqrt(10n-1) to 3n (or, slightly weaker but prettier, in the range from 3*sqrt(n) to 3n).

			a(1)=3 because 3^2+4^2=(5*1)^2; a(2)=6 because 6^2+8^2=(5+2)^2; a(5)=7 since 7^2+24^2=(5*5)^2.

Cf. A036842, A055527.

Offset corrected by Sean A. Irvine, Nov 01 2023

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A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

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A300728 a(n) is the smallest positive number k such that k^2 + k*n + n^2 is a perfect square, or 0 if no such k exists.

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A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

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A238376 Smallest other side of an Heronian triangle with a side of length n.

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A239975 Least k>0 such that n^2 + (n+k)^2 is a square, or -1 if no such k exists.

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A370574 a(n) is the least k > 0 such that n^2 XOR k^2 is a positive square number (where XOR denotes the bitwise XOR operator), or -1 if no such k exists.

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A066662 Shortest leg of a Pythagorean triangle with a hypotenuse of length 5n.

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