A099829 -id:A099829 - cp's OEIS frontend

A099830 Smallest perimeter S such that exactly n distinct Pythagorean triangles with this perimeter can be constructed.

12, 60, 120, 240, 420, 720, 1320, 840, 2640, 1680, 3360, 2520, 4620, 7920, 7560, 5040, 10080, 17160, 10920, 9240, 40320, 25200, 28560, 21840, 18480, 60480, 41580, 46200, 36960, 32760, 27720, 78540, 60060, 129360, 134640, 115920, 85680, 65520, 83160

Offset: 1

Hugo Pfoertner, Oct 27 2004

nonn

Least perimeter common to exactly n distinct Pythagorean triangles. - Lekraj Beedassy, Jun 07 2006

			a(7)=1320 because 1320 is the smallest possible perimeter for which exactly 7 different Pythgorean triangles exist: 1320 = 110+600+610 = 120+594+606 = 220+528+572 = 231+520+569 = 264+495+561 = 330+440+550 = 352+420+548.

Ray Chandler, Table of n, a(n) for n = 1..163
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A099829 first perimeter producing at least n Pythagorean triangles, A009096 ordered perimeters of Pythagorean triangles, A001399, A069905 partitions into 3 parts.

More terms from Ray Chandler, Oct 29 2004

A352881 a(n) is the minimal number z having the largest number of solutions to the Diophantine equation 1/z = 1/x + 1/y such that 1 <= x <= y <= 10^n.

Original entry on oeis.org

2, 12, 60, 840, 9240, 55440, 720720, 6126120, 116396280, 232792560, 5354228880, 26771144400, 465817912560, 4813451763120, 24067258815600, 144403552893600, 2671465728531600, 36510031623265200, 219060189739591200, 4709794079401210800, 18839176317604843200, 221360321731856907600

Offset: 1

Darío Clavijo, Apr 06 2022

nonn

Solving for z gives z = (x*y) / (x+y), so x*y == 0 (mod x+y).
All known terms are from A025487:
  a(1) =    2 = 2;
  a(2) =   12 = 2^2 * 3;
  a(3) =   60 = 2^2 * 3 * 5;
  a(4) =  840 = 2^3 * 3 * 5 * 7;
  a(5) = 9240 = 2^3 * 3 * 5 * 7 * 11.
If a solution to the equation 1/z = 1/x + 1/y is found such that gcd(x,y,z) is a square, then x+y, x*y*z, and (x-y)^2 + (2*z)^2 are also squares.
For all solutions, x^2 + y^2 + z^2 is a square.
The sequence is indeed a subsequence of A025487, and likely of A126098 as well. - Max Alekseyev, Mar 01 2023
a(n) < 5*10^(n-1). - Max Alekseyev, Mar 01 2023

			For n=1, we have the following, where r = (x*y) mod (x+y). (In the last four columns, each number marked by an asterisk is a square.)
.
  r  z  x  y  x*y  x+y  x*y*z  x^2+y^2+z^2
  -  -  -  -  ---  ---  -----  -----------
  0  1  2  2    4*   4*     4*           9* (solution)
  2  1  2  4    8    6      8           21
  4  1  2  6   12    8     12           41
  6  1  2  8   16*  10     16*          69
  3  1  3  3    9    6      9*          19
  0  2  3  6   18*   9*    36*          49* (solution)
  3  2  3  9   27   12     54           94
  0  2  4  4   16*   8     32           36* (solution)
  8  2  4  8   32   12     64*          84
  5  2  5  5   25*  10     50           54
  0  3  6  6   36*  12    108           81* (solution)
  7  3  7  7   49*  14    147          107
  0  4  8  8   64*  16*   256*         144* (solution)
  9  4  9  9   81*  18    324*         178
.
z = 2 has the largest number of solutions, so a(1) = 2.
The number of solutions for the resulting z cannot exceed A018892(z).

Max Alekseyev, Table of n, a(n) for n = 1..30
Project Euler, Diophantine reciprocals I, Problem 108.

Cf. A025487, A099829, A126098, A067128, A018892, A126098, A225206.

a(n)=my(bc=0,bk=0,L=10^n);for(k=1,L-1,my(c=0,k2=k^2);for(d=max(1,k2\(L-k)+1),k,if(k2%d==0,c++););if(c>bc,bc=c;bk=k););return(bk); \\ Darío Clavijo, Mar 03 2025

def a(n):
    # k=x*y and d=x+y
    bc, bk, L = 0, None, 10**n
    for k in range(1, L):
        c, k2 = 0, k * k
        for d in range(max(1, k2 // (L - k) + 1), k + 1):
            if k2 % d == 0: c += 1
        if c > bc:
            bc, bk = c, k
    return bk

a(6) from Chai Wah Wu, Apr 10 2022

a(7)-a(22) from Max Alekseyev, Mar 01 2023

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A099830 Smallest perimeter S such that exactly n distinct Pythagorean triangles with this perimeter can be constructed.

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