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URL: https://oeis.org/A392072

⇱ A392072 - OEIS

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A392072
Smallest integer k such that 1 + x^2 + y^2 = k*prime(n) has at least one solution for some integers x, y.
1
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, 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, 1, 2, 3, 3, 2, 2, 2, 2, 1, 2, 1, 3, 3, 2, 7, 2, 2, 1, 3, 3
OFFSET
1,4
COMMENTS
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.
a(n) is the least k such that k*prime(n)-1 is in A001481. - Robert Israel, Jan 29 2026
LINKS
EXAMPLE
*---*------------*----------------------------*
| n | k | (x,y) | 1 + x^2 + y^2 = k*prime(n) |
*---*------------*----------------------------*
| 1 | 1 | (0,1) | 1 + 0^2 + 1*2 = 1*2 |
| 2 | 1 | (1,1) | 1 + 1^2 + 1^2 = 1*3 |
| 3 | 1 | (0,2) | 1 + 0^2 + 2^2 = 1*5 |
| 4 | 2 | (2,3) | 1 + 2^2 + 3^2 = 2*7 |
| 5 | 1 | (1,3) | 1 + 1^2 + 3^2 = 1*11 |
| 6 | 2 | (0,5) | 1 + 0^2 + 5^2 = 2*13 |
| 7 | 1 | (0,4) | 1 + 0^2 + 4^2 = 1*17 |
| 8 | 1 | (3,3) | 1 + 3^2 + 3^2 = 1*19 |
| 9 | 2 | (3,6) | 1 + 3^2 + 6^2 = 2*23 |
|10 | 5 | (0,12) | 1 + 0^2 + 12^2 =5*29 |
MAPLE
ss:= proc(s)
andmap(t -> t[1] mod 4 <> 3 or t[2]::even, ifactors(s)[2])
end proc:
f:= proc(n) local p, k;
p:= ithprime(n);
for k from 1 do if ss(k*p-1) then return k fi od;
end proc:
map(f, [$1..100]); # Robert Israel, Jan 29 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 29 2026
STATUS
approved