A053761 Least positive integer k for which the Jacobi symbol (k|2*n-1) is less than 1, where 2*n-1 is a nonsquare; a(n)=0 if 2*n-1 is a square.

0, 2, 2, 3, 0, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3, 0, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 7, 5, 2, 2, 3

Offset: 1

Author

Steven Finch, Apr 05 2000

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
• Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., Springer-Verlag 1996; Math. Rev. 96k:11112.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• R. Baillie and S. S. Wagstaff, Lucas pseudoprimes, Math. Comp. 35 (1980) 1391-1417; Math. Rev. 81j:10005
• Steven R. Finch, Quadratic Residues [Broken link]
• Steven R. Finch, Quadratic Residues [From the Wayback machine]

Crossrefs

Cf. A010052, A053760, A112046, A112049, A268829.

Programs

Maple

A053761 := proc(n) if issqr(2*n-1) then return 0 ; else for k from 1 do if numtheory[jacobi](k,2*n-1) < 1 then return k; end if; end do: end if; end proc: seq(A053761(n),n=1..100) ; # R. J. Mathar, Aug 08 2010

Mathematica

a[n_] := If[IntegerQ[Sqrt[2*n - 1]], Return[0], For[ k = 1, True, k++, If[ JacobiSymbol[k, 2*n - 1] < 1 , Return[k]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 20 2013, after R. J. Mathar *)

PARI

A112046(n) = for(i=1,(2*n),if((kronecker(i,(n+n+1)) < 1),return(i)));
A053761(n) = if(issquare((2*n)-1),0,A112046(n-1));
for(n=1, 10000, write("b053761.txt", n, " ", A053761(n))); \\ Antti Karttunen, May 10 2017

Scheme

(define (A053761 n) (if (= 1 n) 0 (* (- 1 (A010052 (+ n n -1))) (A112046 (- n 1))))) ;; Antti Karttunen, May 10 2017

Formula

a(1) = 0; for n > 1, a(n) = (1-A010052((2*n)-1)) * A112046(n-1). - Antti Karttunen, May 10 2017

Extensions

More terms from R. J. Mathar, Aug 08 2010