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 A080909 a(n) = (2n+1)! modulo 4n+3, |a(n)| <= 1. 0
 1, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, -1, 0, 0, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, -1, 0, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS If 4n+3 is composite, then a(n)=0. If 4n+3 is prime, then a(n)=(-1)^m where m is the number of quadratic non-residues less than or equal to 2n+1. Is there a way to predict whether a(n)=1 or a(n)=-1? REFERENCES G. H. Hardy and E. M. Wright, An introduction to the theory of number, fourth edition, 1960, section 7.7: the residue of ((p-1)/2)!. LINKS Table of n, a(n) for n=0..82. FORMULA a(n) = mods((2*n+1)!, 4*n+3). EXAMPLE a(3) = 0 since 7! == 0 (mod 15). a(4) = 1 since 9! == -1 (mod 19). MAPLE seq(mods((2*n+1)!, 4*n+3), n=0..100); MATHEMATICA a[ n_] := Mod[(2*n+1)!, 4*n+3, -1]; (* Michael Somos, Jul 25 2023 *) PROG (PARI) a(n)= {v =(2*n+1)! % (4*n+3); if (2*v > 4*n+3, v -= 4*n+3); return (v); } \\ Michel Marcus, Jul 21 2013 CROSSREFS Sequence in context: A267533 A118274 A275737 * A087755 A050072 A267576 Adjacent sequences: A080906 A080907 A080908 * A080910 A080911 A080912 KEYWORD sign AUTHOR Christophe Leuridan (ChristopheLeuridan(AT)ujf-grenoble.fr), Apr 01 2003 EXTENSIONS More terms from Michel Marcus, Jul 21 2013 Name edited by Michael Somos, Jul 25 2023 STATUS approved

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Last modified June 20 23:53 EDT 2024. Contains 373535 sequences. (Running on oeis4.)