A192453 Numbers k such that -1 is a 4th power mod k.

1, 2, 17, 34, 41, 73, 82, 89, 97, 113, 137, 146, 178, 193, 194, 226, 233, 241, 257, 274, 281, 289, 313, 337, 353, 386, 401, 409, 433, 449, 457, 466, 482, 514, 521, 562, 569, 577, 578, 593, 601, 617, 626, 641, 673, 674, 697, 706, 761, 769, 802, 809, 818, 857

Offset: 1

Author

José María Grau Ribas, Jul 01 2011

Keywords

• nonn

Comments

Complement of A192452. Subsequence of A008784. A further reduction to 8th powers yields 1, 2, 17, 34, 97, 113, 193, 194, ...

From Jianing Song, Mar 31 2019: (Start)

k is a term if and only if k is not divisible by 4 and all odd prime factors are congruent to 1 modulo 8. If k is a term of this sequence, then so are all divisors of k.

Decompose the multiplicative group of integers modulo k as a product of cyclic groups C_{s_1} x C_{s_2} x ... x C_{s_m}, where s_i divides s_j for i < j, then k is a term iff s_1 is divisible by 8. For k = 1 or 2, (Z/kZ)* is the trivial group, s_1 does not exist so 1 and 2 are also terms. This is an analog of A008784 (where s_1 is divisible by 4) and A319100 (where s_1 is divisible by 6). (End)

Examples

			1^4 == -1 (mod 1). 2^4 == -1 (mod 17). 9^4 == -1 (mod 34). 3^4 == -1 (mod 41). 10^4 == -1 (mod 73).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..4380 (all terms below 10^5)

Crossrefs

Cf. A008784, A047232, A319100.

Programs

Maple

select(n -> numtheory:-factorset(n) mod 8 subset {1,2}, [seq(seq(4*i+j,j=1..3),i=0..400)]); # Robert Israel, May 24 2019

Mathematica

Table[If[Reduce[x^4==-1,Modulus->n]===False,Null,n],{n,2,1000}]//Union

PARI

for(n=1,1e3,if(ispower(Mod(-1,n),4),print1(n", "))) \\ Charles R Greathouse IV, Jul 03 2011