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.

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

Offset: 1

Author

Darío Clavijo, Apr 06 2022

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

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

Crossrefs

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

Programs

PARI

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

Python

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

Extensions

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

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