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A192453
Numbers k such that -1 is a 4th power mod k.
5
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
1,2
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)
LINKS
Jianing Song, Table of n, a(n) for n = 1..4380 (all terms below 10^5)
EXAMPLE
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).
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
PROG
(PARI) for(n=1, 1e3, if(ispower(Mod(-1, n), 4), print1(n", "))) \\ Charles R Greathouse IV, Jul 03, 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved