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A059287 Primes p such that x^16 = 2 has no solution mod p, but x^8 = 2 has a solution mod p.

%I A059287 #24 Jun 23 2024 11:26:40
%S A059287 1217,1249,1553,1777,2833,4049,4273,4481,4993,5297,6449,6481,6689,
%T A059287 7121,8081,8609,9137,9281,9649,10337,10369,10433,11329,11617,11633,
%U A059287 12241,12577,13121,13441,13633,14321,14753,15121,15569,16417,16433,16673
%N A059287 Primes p such that x^16 = 2 has no solution mod p, but x^8 = 2 has a solution mod p.
%H A059287 Vincenzo Librandi, <a href="/A059287/b059287.txt">Table of n, a(n) for n = 1..150</a>
%t A059287 Select[Prime[Range[PrimePi[20000]]], !MemberQ[PowerMod[Range[#], 16, #], Mod[2, #]] && MemberQ[PowerMod[Range[#], 8, #], Mod[2, #]]&] (* _Vincenzo Librandi_, Sep 21 2013 *)
%o A059287 (Magma) [p: p in PrimesUpTo(17000) | not exists{x: x in ResidueClassRing(p) | x^16 eq 2} and exists{x: x in ResidueClassRing(p) | x^8 eq 2}]; // _Vincenzo Librandi_, Sep 21 2012
%o A059287 (PARI) select( {is_A059287(p)=Mod(2,p)^(p\gcd(8,p-1))==1&&Mod(2,p)^(p\gcd(16,p-1))!=1}, primes(1999)) \\ Could any composite number pass this test? - _M. F. Hasler_, Jun 22 2024
%o A059287 (Python)
%o A059287 from itertools import islice
%o A059287 from sympy import is_nthpow_residue, nextprime
%o A059287 def A059287_gen(startvalue=2): # generator of terms >= startvalue
%o A059287     p = max(1,startvalue-1)
%o A059287     while (p:=nextprime(p)):
%o A059287         if is_nthpow_residue(2,8,p) and not is_nthpow_residue(2,16,p):
%o A059287             yield p
%o A059287 A059287_list = list(islice(A059287_gen(),10)) # _Chai Wah Wu_, Jun 23 2024
%Y A059287 Cf. A000040, A045315, A045316.
%Y A059287 Cf. A070184 (same with x^64 instead of x^16).
%K A059287 nonn,easy
%O A059287 1,1
%A A059287 _Klaus Brockhaus_, Jan 25 2001