A357408 a(n) is the least sum n + y such that 1/n + 1/y = 1/z with gcd(n,y,z) = 1, for some integers y and z.

4, 9, 16, 25, 9, 49, 64, 81, 25, 121, 16, 169, 49, 25, 256, 289, 81, 361, 25, 49, 121, 529, 64, 625, 169, 729, 49, 841, 36, 961, 1024, 121, 289, 49, 81, 1369, 361, 169, 64, 1681, 49, 1849, 121, 81, 529, 2209, 256, 2401, 625, 289, 169, 2809, 729, 121, 64, 361

Offset

2,1

Comments

All terms are squares.

Proof: Consider the general equation 1/x + 1/y = 1/z, x,y,z positive integers and gcd(x,y,z) = 1. If x,y,z is a solution, let a = x-z and b = y-z. Then a*b = (x-z)*(y-z) = z^2. The equation 1/x + 1/y = 1/z describes a projective curve in 2-dimensional projective space P^2, given by the homogeneous equation x*y - x*z - y*z = 0. So the solution is of the form x = a + sqrt(a*b), y = b + sqrt(a*b), z = sqrt(a*b), for all positive integers a and b where a*b is a square. The solutions with gcd(x,y,z) = 1 are exactly x = c^2 + c*d, y = d^2 + c*d, z = c*d for coprime positive integers c and d. Therefore the sum x+y = (c+d)^2 is a square, and x-z, y-z are also squares.

Links

Table of n, a(n) for n=2..57.

Example

a(3) = 9 because 1/3 + 1/6 = 1/2 with 3 + 6 = 9.
Table with the integers n, y, z, n+y, c and d, for n >= 2:
  +-----+------+-----+-------------+-----+-----+
  |  n  |   y  |  z  |  a(n) = n+y |  c  |  d  |
  +-----+------+-----+-------------+-----+-----+
  |  2  |   2  |  1  |       4     |  1  |  1  |
  |  3  |   6  |  2  |       9     |  1  |  2  |
  |  4  |  12  |  3  |      16     |  1  |  3  |
  |  5  |  20  |  4  |      25     |  1  |  4  |
  |  6  |   3  |  2  |       9     |  2  |  1  |
  |  7  |  42  |  6  |      49     |  1  |  6  |
  |  8  |  56  |  7  |      64     |  1  |  7  |
  |  9  |  72  |  8  |      81     |  1  |  8  |
  | 10  |  15  |  6  |      25     |  2  |  3  |
  | 11  | 110  | 10  |     121     |  1  | 10  |
  | 12  |   4  |  3  |      16     |  3  |  1  |
  | 13  | 156  | 12  |     169     |  1  | 12  |
  | 14  |  35  | 10  |      49     |  2  |  5  |
  | 15  |  10  |  6  |      25     |  3  |  2  |
  | 16  | 240  | 15  |     256     |  1  | 15  |
  | 17  | 272  | 16  |     289     |  1  | 16  |
  | 18  |  63  | 14  |      81     |  2  |  7  |
  | 19  | 342  | 18  |     361     |  1  | 18  |
  | 20  |   5  |  4  |      25     |  4  |  1  |

Maple

nn:=3000:T:=array(1..56):
for n from 2 to 57 do:
ii:=0:
for y from 1 to nn while(ii=0)do:
   x1:=evalf(n*y/(n+y)):y1:=floor(x1):
   g1:=gcd(n, y):g2:=gcd(g1, y1):
    if x1=y1 and g2=1
     then
     printf(`%d %d %d %d\n`, n, y, y1, n+y):ii:=1:T[n-1]:=n+y:
     else fi:
   od:
  od:
print(T):

Crossrefs

Cf. A000290 (squares), A034699.

Sequence in context: A070451 A070450 A070449 * A070448 A081403 A259602

Adjacent sequences: A357405 A357406 A357407 * A357409 A357410 A357411

Keyword

nonn

Author

Michel Lagneau, Sep 26 2022

Status

approved