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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A265991 A322871

Adjacent sequences:  A316972 A316973 A316974 * A316976 A316977 A316978

KEYWORD

nonn,mult

AUTHOR

Arnauld Chevallier, Jul 17 2018

STATUS

approved

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Last modified March 1 03:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)