A163366 a(n) = (-1)^floor((prime(n)+2)/2) mod prime(n).

1, 1, 4, 1, 1, 12, 16, 1, 1, 28, 1, 36, 40, 1, 1, 52, 1, 60, 1, 1, 72, 1, 1, 88, 96, 100, 1, 1, 108, 112, 1, 1, 136, 1, 148, 1, 156, 1, 1, 172, 1, 180, 1, 192, 196, 1, 1, 1, 1, 228, 232, 1, 240, 1, 256, 1, 268, 1, 276, 280, 1, 292, 1, 1, 312, 316, 1, 336, 1, 348, 352, 1, 1, 372, 1

Offset

1,3

Comments

Remove the '1's from the sequence to get A152680.

Product modulo p of the quadratic residues of p, where p = prime(n). [Jonathan Sondow, May 14 2010]

References

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398. [Jonathan Sondow, May 14 2010]

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Rahul Gupta, Algorithmic Number Theory, Section 24.5 [Jonathan Sondow, May 14 2010]

Formula

a(n)*A177863(n) == -1 (mod prime(n)), by Wilson's theorem. - Jonathan Sondow, May 14 2010

a(n) = A177860(n) modulo prime(n). - Jonathan Sondow, May 14 2010

Example

a(4) = 1 because the quadratic residues of prime(4) = 7 are 1, 2, and 4, and 1*2*4 = 8 == 1 (mod 7). - Jonathan Sondow, May 14 2010

Maple

seq((-1)^iquo(ithprime(i)+2, 2) mod ithprime(i), i=1..113);

Mathematica

Table[Mod[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], Prime[n]], {n, 1, 80}] (* Jonathan Sondow, May 14 2010 *)

Crossrefs

Cf. A152680, A005098, A002144, A009003.

Cf. A177860, A177863. - Jonathan Sondow, May 14 2010

Sequence in context: A220688 A146990 A051433 * A181145 A227203 A140070

Adjacent sequences: A163363 A163364 A163365 * A163367 A163368 A163369

Keyword

nonn

Author

Peter Luschny, Jul 25 2009

Status

approved