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A048280 Length of longest run of consecutive quadratic residues mod prime(n). 3
2, 2, 3, 3, 3, 3, 5, 4, 5, 4, 4, 4, 5, 5, 5, 3, 5, 5, 6, 7, 9, 6, 7, 5, 9, 7, 7, 6, 5, 5, 7, 8, 6, 5, 4, 7, 6, 6, 6, 6, 6, 6, 7, 9, 7, 6, 7, 7, 7, 5, 6, 7, 13, 7, 6, 7, 8, 7, 10, 6, 9, 9, 7, 11, 9, 5, 8, 9, 8, 6, 6, 8, 9, 6, 8, 8, 8, 5, 7, 13, 8, 7, 7, 9, 10, 8, 8, 9, 8, 8, 11, 13, 8, 8, 10, 8, 9, 8, 10, 7, 9, 9, 10, 10, 7, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

0 and 1 are consecutive quadratic residues for any prime, so a(n) >= 2.

"Consecutive" allows wrap-around, so p-1 and 0 are consecutive. - Robert Israel, Jul 20 2014

A002307(n) is defined similarly, except that only positive reduced quadratic residues are counted. - Jonathan Sondow, Jul 20 2014

For longest runs of quadratic nonresidues, see A002308. - Jonathan Sondow, Jul 20 2014

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

P. Pollack and E. Treviño, The primes that Euclid forgot, Amer. Math. Monthly, 121 (2014), 433-437.

Enrique Treviño, Corrigendum to “On the maximum number of consecutive integers on which a character is constant”, Mosc. J. Comb. Number Theory 7 (2017), no. 3, 1-2.

FORMULA

a(n) < 2*sqrt(prime(n)) for n >= 1 (see Pollack and Treviño for n > 1). - Jonathan Sondow, Jul 20 2014

a(n) >= A002307(n). - Jonathan Sondow, Jul 20 2014

a(n) < 7 prime(n)^(1/4)log(prime(n)) for all n > 1, or a(n) < 3.2 prime(n)^(1/4)log(prime(n)) for n >= 10^13. - Enrique Treviño, Apr 16 2020

EXAMPLE

For n = 7, prime(7) = 17 has consecutive quadratic residues 15,16,0,1,2, and no longer sequence of consecutive quadratic residues, so a(7)=5.

MAPLE

A:= proc(n) local P, res, nonres, nnr;

     P:= ithprime(n);

     res:= {seq(i^2, i=0..floor((P-1)/2)};

     nonres:= {$1..P-1} minus res;

     nnr:= nops(nonres);

     max(seq(nonres[i+1]-nonres[i]-1, i=1..nnr-1), nonres[1]-nonres[-1]+P-1)

end proc;

A(1):= 2:

seq(A(n), n=1..100); # Robert Israel, Jul 20 2014

CROSSREFS

Cf. A002307, A048281.

Sequence in context: A034584 A035430 A167227 * A024695 A259195 A143997

Adjacent sequences:  A048277 A048278 A048279 * A048281 A048282 A048283

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

Offset corrected to 1 and definition clarified by Jonathan Sondow Jul 20 2014

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)