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%I #54 Nov 03 2023 08:04:35
%S 23,59,101,167,173,211,223,271,307,317,347,449,463,593,599,607,691,
%T 719,809,821,829,853,877,883,991,997,1097,1117,1151,1163,1181,1231,
%U 1319,1451,1453,1481,1553,1613,1669,1697,1787,1789,1867,1871,1879,1889,1913,2027,2053,2143,2309,2339,2347,2381,2393,2423,2539,2647,2677,2693,2707,2741,2819
%N Primes of form x^2 + 23*y^2.
%C Discriminant -23.
%C Also primes of the form x^2 + x*y + 6*y^2. - _N. J. A. Sloane_, Jun 02 2014
%C Also primes of the form x^2 - x*y + 6*y^2 with x and y nonnegative. - _T. D. Noe_, May 07 2005
%C Primes p such that X^3-X+1 is split modulo p. E.g., X^3-X+1 = (X-33)*(X-40)*(X-94) modulo 167. - Julien Freslon (julien.freslon(AT)wanadoo.fr), Feb 24 2007
%C It appears that, if x > 0, then tau(p) = A000594(p) == 2 (mod 23). - Comment from _Jud McCranie_
%C In fact, this sequence appears to be the same as primes p such that RamanujanTau(p) == {1,2} (mod 23). - _Ray Chandler_, Dec 01 2016
%C Excluding the first term, this sequence is the intersection of A191021 and A256567. - _Arkadiusz Wesolowski_, Oct 03 2021
%D David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
%D Joe Roberts, Lure of the Integers, "Integer 23 - the Tau function".
%H Ray Chandler, <a href="/A033217/b033217.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H John Raymond Wilton, <a href="https://doi.org/10.1112/plms/s2-31.1.1">Congruence properties of Ramanujan's function τ(n)</a>, Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.
%H J. R. Wilton, <a href="/A278578/a278578.pdf">Congruence properties of Ramanujan's function τ(n)</a>, annotated copy of page 8 only.
%t QuadPrimes2[1, 0, 23, 10000] (* see A106856 *)
%t Join[{23}, nn=23; pMax=5000; Union[Reap[Do[p=x^2 + nn y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]]] (* _Vincenzo Librandi_, Sep 05 2016 *)
%o (PARI) isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ _Arkadiusz Wesolowski_, Oct 03 2021
%o (PARI) isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ _Arkadiusz Wesolowski_, Oct 09 2021
%Y Cf. A000594, A191021, A256567. Primes in A028958.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
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