%I #39 Sep 08 2022 08:45:02

%S 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,

%T 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,179,181,

%U 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277

%N Primes p such that x^43 = 2 has a solution mod p.

%C Primes not of the form 43k + 1. - _Charles R Greathouse IV_, Aug 22 2011 [Not so! The smallest counterexample is 5419: 5419 = 43*126 + 1, but 2^43 == 2 (mod 5419), so it is here. - _Jianing Song_, Mar 07 2021]

%C Differs from A000040 - the prime 173 does not appear.

%C For case x^31 = 2 the first missing prime is 311 (64th term).

%C For case x^47 = 2 the first missing prime is 283 (61st term).

%C For case x^59 = 2 the first missing prime is 709 (127th term).

%C For case x^61 = 2 the first missing prime is 367 (73rd term).

%C Complement of A059243 relative to A000040. - _Vincenzo Librandi_, Sep 14 2012

%C From _Jianing Song_, Mar 07 2021: (Start)

%C It is conjectured that this sequence has density 42/43 ~ 0.976744 over all the primes.

%C N | # of terms among

%C | the first N primes

%C ------+--------------------

%C 10^4 | 9758

%C 10^5 | 97681

%C 10^6 | 976798

%C 10^7 | 9767551

%C 10^8 | 97674723

%C If the conjecture is correct, then a(n) ~ 43/42 * n log n.

%C In general, let p be a prime, a be an integer that is not a p-th power, then it seems that the density of prime factors of x^p - a over all the primes is 1 - 1/p. This is well-known to be correct for p = 2. (End)

%C The generalized conjecture above is equivalent to: let P(p,1) be the set of primes congruent to 1 modulo p, P(p,1;a) be the set of primes q congruent to 1 modulo p such that x^p == a (mod q) has a solution, where p is a prime, a is not a p-th power, then the density of P(p,1;a) over P(p,1) is 1/p. - _Jianing Song_, Mar 09 2021

%H R. J. Mathar, <a href="/A058853/b058853.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Pri#smp">Index entries for related sequences</a>

%t ok[p_]:= Reduce[Mod[x^43 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[1000]], ok] (* _Vincenzo Librandi_ Sep 14 2012 *)

%o (Magma) [p: p in PrimesUpTo(400) | exists(t){x : x in ResidueClassRing(p) | x^43 eq 2}]; // _Vincenzo Librandi_ Sep 14 2012

%o (PARI) isA058853(p) = isprime(p) && ispower(Mod(2,p), 43) \\ _Jianing Song_, Mar 07 2021

%K nonn,easy

%O 1,1

%A _Patrick De Geest_, Dec 15 2000

%E The old formula "a(n) ~ 42/41 * n log n" based on false observation from _Charles R Greathouse IV_, Aug 22 2011 removed by _Jianing Song_, Mar 07 2021