A357408 - OEIS

%I #23 Nov 06 2022 08:38:25

%S 4,9,16,25,9,49,64,81,25,121,16,169,49,25,256,289,81,361,25,49,121,

%T 529,64,625,169,729,49,841,36,961,1024,121,289,49,81,1369,361,169,64,

%U 1681,49,1849,121,81,529,2209,256,2401,625,289,169,2809,729,121,64,361

%N 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.

%C All terms are squares.

%C 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.

%e a(3) = 9 because 1/3 + 1/6 = 1/2 with 3 + 6 = 9.

%e Table with the integers n, y, z, n+y, c and d, for n >= 2:

%e   +-----+------+-----+-------------+-----+-----+

%e   |  n  |   y  |  z  |  a(n) = n+y |  c  |  d  |

%e   +-----+------+-----+-------------+-----+-----+

%e   |  2  |   2  |  1  |       4     |  1  |  1  |

%e   |  3  |   6  |  2  |       9     |  1  |  2  |

%e   |  4  |  12  |  3  |      16     |  1  |  3  |

%e   |  5  |  20  |  4  |      25     |  1  |  4  |

%e   |  6  |   3  |  2  |       9     |  2  |  1  |

%e   |  7  |  42  |  6  |      49     |  1  |  6  |

%e   |  8  |  56  |  7  |      64     |  1  |  7  |

%e   |  9  |  72  |  8  |      81     |  1  |  8  |

%e   | 10  |  15  |  6  |      25     |  2  |  3  |

%e   | 11  | 110  | 10  |     121     |  1  | 10  |

%e   | 12  |   4  |  3  |      16     |  3  |  1  |

%e   | 13  | 156  | 12  |     169     |  1  | 12  |

%e   | 14  |  35  | 10  |      49     |  2  |  5  |

%e   | 15  |  10  |  6  |      25     |  3  |  2  |

%e   | 16  | 240  | 15  |     256     |  1  | 15  |

%e   | 17  | 272  | 16  |     289     |  1  | 16  |

%e   | 18  |  63  | 14  |      81     |  2  |  7  |

%e   | 19  | 342  | 18  |     361     |  1  | 18  |

%e   | 20  |   5  |  4  |      25     |  4  |  1  |

%p nn:=3000:T:=array(1..56):

%p for n from 2 to 57 do:

%p ii:=0:

%p for y from 1 to nn while(ii=0)do:

%p    x1:=evalf(n*y/(n+y)):y1:=floor(x1):

%p    g1:=gcd(n,y):g2:=gcd(g1,y1):

%p     if x1=y1 and g2=1

%p      then

%p      printf(`%d %d %d %d\n`,n,y,y1,n+y):ii:=1:T[n-1]:=n+y:

%p      else fi:

%p    od:

%p   od:

%p print(T):

%Y Cf. A000290 (squares), A034699.

%K nonn

%O 2,1

%A _Michel Lagneau_, Sep 26 2022