A392072 - OEIS

%I #49 Feb 04 2026 15:54:46

%S 1,1,1,2,1,2,1,1,2,5,2,1,1,2,5,1,1,2,3,3,1,2,1,6,2,1,2,1,5,2,5,1,1,2,

%T 1,3,2,1,2,6,1,1,5,3,1,2,2,2,1,2,1,2,2,1,1,3,6,2,5,3,2,1,1,3,2,6,2,2,

%U 1,2,3,3,2,2,2,2,1,2,1,3,3,2,7,2,2,1,3,3

%N Smallest integer k such that 1 + x^2 + y^2 = k*prime(n) has at least one solution for some integers x, y.

%C We use a classic property of the sum of three squares: for any odd prime number p, there exist integers x, y, z such that 1 + x^2 + y^2 = z*p with 0 < z < p.

%C a(n) is the least k such that k*prime(n)-1 is in A001481. - _Robert Israel_, Jan 29 2026

%H Robert Israel, <a href="/A392072/b392072.txt">Table of n, a(n) for n = 1..10000</a>

%e *---*------------*----------------------------*

%e | n | k | (x,y)  | 1 + x^2 + y^2 = k*prime(n) |

%e *---*------------*----------------------------*

%e | 1 | 1 | (0,1)  |  1 + 0^2 + 1*2 = 1*2       |

%e | 2 | 1 | (1,1)  |  1 + 1^2 + 1^2 = 1*3       |

%e | 3 | 1 | (0,2)  |  1 + 0^2 + 2^2 = 1*5       |

%e | 4 | 2 | (2,3)  |  1 + 2^2 + 3^2 = 2*7       |

%e | 5 | 1 | (1,3)  |  1 + 1^2 + 3^2 = 1*11      |

%e | 6 | 2 | (0,5)  |  1 + 0^2 + 5^2 = 2*13      |

%e | 7 | 1 | (0,4)  |  1 + 0^2 + 4^2 = 1*17      |

%e | 8 | 1 | (3,3)  |  1 + 3^2 + 3^2 = 1*19      |

%e | 9 | 2 | (3,6)  |  1 + 3^2 + 6^2 = 2*23      |

%e |10 | 5 | (0,12) |  1 + 0^2 + 12^2 =5*29      |

%p ss:= proc(s)

%p   andmap(t -> t[1] mod 4 <> 3 or t[2]::even, ifactors(s)[2])

%p end proc:

%p f:= proc(n) local p,k;

%p       p:= ithprime(n);

%p       for k from 1 do if ss(k*p-1) then return k fi od;

%p end proc:

%p map(f,[$1..100]); # _Robert Israel_, Jan 29 2026

%Y Cf. A000378, A001481, A133529.

%K nonn

%O 1,4

%A _Michel Lagneau_, Jan 29 2026