A295784 Length of the longest arithmetic progression in squares mod n with slope coprime to n.
2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 4, 3, 2, 2, 5, 2, 4, 2, 2, 3, 5, 2, 3, 3, 2, 2, 4, 2, 4, 2, 2, 5, 3, 2, 4, 4, 2, 2, 5, 2, 5, 2, 2, 5, 5, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 4, 5, 2, 5, 4, 2, 2, 3, 2, 6, 2, 2, 3, 7, 2, 9, 4, 2, 2, 3, 2, 6, 2, 2, 5, 7, 2, 3, 5, 2, 2, 5, 2, 3, 2, 2, 5, 3, 2, 9, 3, 2, 2, 7, 2, 7, 2, 2, 6, 6
Offset: 3
Keywords
Examples
For n=17 we have residues {0,1,2,4,8,9,13,15,16} and the following arithmetic progressions of length 5: (15, 16, 0, 1, 2), (13, 15, 0, 2, 4), (9, 13, 0, 4, 8)
Links
- Tom Hejda, Table of n, a(n) for n = 3..20000
- MathOverflow user Seva, Consecutive non-quadratic residues (Proves that a(n) is unbounded.)
Programs
-
SageMath
def a(n) : if n in [1,2] : return Infinity R = quadratic_residues(n) return max( next( m for m in itertools.count() if (a+(b-a)*m)%n not in R ) \ for a,b in zip(R,R[1:]+R[:1]) if gcd(b-a,n) == 1 )
Comments