A002307 Consecutive quadratic residues mod p: a(n) is the maximal number of positive reduced quadratic residues which appear consecutively for n-th prime.

1, 1, 1, 2, 3, 2, 2, 4, 4, 4, 4, 4, 3, 5, 4, 3, 5, 5, 6, 6, 4, 6, 7, 4, 4, 7, 7, 6, 5, 5, 7, 8, 6, 5, 4, 7, 6, 6, 6, 6, 6, 6, 6, 4, 7, 6, 7, 7, 7, 5, 6, 6, 6, 7, 6, 7, 8, 7, 10, 6, 9, 9, 7, 10, 5, 5, 8, 5, 8, 6, 6, 8, 9, 6, 8, 8, 8, 5, 7, 6, 8, 7, 6, 7, 10, 8, 8, 5, 8, 8, 11, 12, 8, 8, 10, 8, 9, 8, 10, 7, 9, 9, 10, 10, 7, 6, 9

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

When prime(n) == 3 (mod 4), then a(n) = A002308(n). - T. D. Noe, Apr 03 2007

A048280(n) is defined similarly, except that reduced quadratic residues equal to 0 are also included. - Jonathan Sondow, Jul 20 2014

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• A. A. Bennett, Consecutive quadratic residues, Bull. Amer. Math. Soc., 32 (1926), 283-284.

Crossrefs

Cf. A002308, A048280, A097159.

Programs

Mathematica

f[l_, a_] := Module[{A = Split[l], B}, B = Last[ Sort[ Cases[A, x : {a ..} :> {Length[x], Position[A, x][[1, 1]]}]]]; {First[B], Length[ Flatten[ Take[A, Last[B] - 1]]] + 1}]; g[n_] := f[ JacobiSymbol[ Range[ Prime[n] - 1], Prime[n]], 1][[1]]; Table[ g[n], {n, 2, 102}] (* Robert G. Wilson v, Jul 28 2004 *)

Formula

a(n) <= A048280(n) < 2*sqrt(prime(n)). - Jonathan Sondow, Jul 20 2014

Extensions

More terms from David W. Wilson