A232986 Numbers m == 1 (mod 4) such that the Gauss factorial ((m-1)/4, m)! == 1 (mod m).

5, 145, 205, 725, 1025, 1105, 1145, 1205, 1305, 1313, 1365, 1405, 1469, 1745, 1785, 1845, 1885, 1989, 2145, 2249, 2405, 2465, 2545, 2665, 2745, 2805, 3005, 3045, 3145, 3161, 3205, 3393, 3445, 3485, 3545, 3601, 3625, 3705, 3885, 3893, 3965

Offset

1,1

Comments

The Gauss factorial m_k! is defined to be Product_{1<=j<=m, gcd(j,k)=1} j.

Links

Table of n, a(n) for n=1..41.

J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810-828.

J. B. Cosgrave and K. Dilcher, The Gauss-Wilson theorem for quarter-intervals, Acta Mathematica Hungarica, Sept. 2013.

Example

m=145 is a term, because 36_145! = 32577412307818387955599294857216 == 1 (mod 145).

Maple

Gf:=proc(N, n) local j, k; k:=1;
for j from 1 to N do if gcd(j, n)=1 then k:=j*k; fi; od; k; end;
t1:=[];
for i from 1 to 1000 do
n:=4*i+1; if (Gf(i, n) mod n ) = 1 then t1:=[op(t1), n]; fi;
od:
t1;

Crossrefs

Sequence in context: A320414 A168041 A081322 * A322954 A254711 A273920

Adjacent sequences: A232983 A232984 A232985 * A232987 A232988 A232989

Keyword

nonn

Author

N. J. A. Sloane, Dec 08 2013

Status

approved