A316975 a(n) is the minimum number of occurrences of the same value (r0-r1+n) mod n for all pairs (r0,r1) of quadratic residues mod n.

1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 4, 1, 1, 4, 2, 5, 1, 2, 6, 6, 1, 2, 6, 2, 2, 7, 2, 8, 1, 3, 8, 2, 1, 9, 10, 3, 1, 10, 4, 11, 3, 1, 12, 12, 1, 8, 4, 4, 3, 13, 4, 3, 2, 5, 14, 15, 1, 15, 16, 2, 1, 3, 6, 17, 4, 6, 4, 18, 1, 18, 18, 2, 5, 6, 6, 20, 1, 4, 20

Offset

1,2

Comments

Multiplicative by the Chinese remainder theorem since if gcd(m,n) = 1 then r is a quadratic residue mod m*n iff it is a quadratic residue mod m and a quadratic residue mod n. - Andrew Howroyd, Aug 07 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Example

The quadratic residues mod 10 are 0, 1, 4, 5, 6 and 9. For each pair (r0,r1) of these quadratic residues, we compute (r0-r1+10) mod 10, leading to:
      0 1 4 5 6 9
    +------------
  0 | 0 9 6 5 4 1
  1 | 1 0 7 6 5 2
  4 | 4 3 0 9 8 5
  5 | 5 4 1 0 9 6
  6 | 6 5 2 1 0 7
  9 | 9 8 5 4 3 0
The minimum number of occurrences of the same value in the above table is 2 (for 2, 3, 7 and 8). Therefore a(10) = 2.

Mathematica

Table[Min@ Tally[#][[All, -1]] &@ Flatten[Mod[#, n] & /@ Outer[Subtract, #, #]] &@ Union@ PowerMod[Range@ n, 2, n], {n, 82}] (* Michael De Vlieger, Jul 20 2018 *)

Prog

(PARI) a(n)={my(r=Set(vector(n, i, i^2%n))); my(v=vector(n)); for(i=1, #r, for(j=1, #r, v[(r[i]-r[j])%n + 1]++)); vecmin(select(t->t>0, v))} \\ Andrew Howroyd, Aug 07 2018

Crossrefs

Cf. A096008.

Sequence in context: A165190 A025890 A334440 * A043277 A379018 A265991

Adjacent sequences: A316972 A316973 A316974 * A316976 A316977 A316978

Keyword

nonn,mult

Author

Arnauld Chevallier, Jul 17 2018

Status

approved