A177860 -id:A177860 - cp's OEIS frontend

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

Peter Luschny, Jul 25 2009

nonn

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]

			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

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

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]

Cf. A152680, A005098, A002144, A009003.

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

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

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 *)

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

A177861 Product of the quadratic nonresidues of prime(n).

Original entry on oeis.org

1, 2, 6, 90, 6720, 36960, 11642400, 283046400, 2412984420000, 1140422816332800, 1226781977195174400, 1863152400854384640000, 5988092802221559085056000, 112886540292742916603904000, 158983195607776600998537600000000

Offset: 1

Jonathan Sondow, May 14 2010

a(n) == (-1)^((p-1)/2) (mod p), if p = prime(n) is odd.

			The quadratic nonresidues of prime(4) = 7 are 3, 5, and 6, so a(4) = 3*5*6 = 90.

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398.

Rahul Gupta, Algorithmic Number Theory, Section 24.5

A125615 Sum of the quadratic nonresidues of prime(n), A177860 Product of the quadratic residues of prime(n), A177863 Product of the quadratic nonresidues of prime(n) modulo prime(n).

Table[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], -1]]], {n, 1, 16}]

a(n) = (p-1)!/A177860(n), where p = prime(n).

cp's OEIS Frontend

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

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