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 A033217 Primes of form x^2 + 23*y^2. 6
 23, 59, 101, 167, 173, 211, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 829, 853, 877, 883, 991, 997, 1097, 1117, 1151, 1163, 1181, 1231, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Discriminant -23. Also primes of the form x^2 + xy + 6y^2. - N. J. A. Sloane, Jun 02 2014 Also primes of the form x^2 - xy + 6y^2 with x and y nonnegative. - T. D. Noe, May 07 2005 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 It appears that, if x>0, then tau(p) = A000594(p) == 2 mod 23. - Comment from Jud McCranie 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 Excluding the first term, this sequence is the intersection of A191021 and A256567. - Arkadiusz Wesolowski, Oct 03 2021 REFERENCES David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989. Joe Roberts, Lure of the Integers, "Integer 23 - the Tau function". LINKS T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe] N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) John Raymond Wilton, Congruence properties of Ramanujan's function τ(n), Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II. J. R. Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only. MATHEMATICA QuadPrimes2[1, 0, 23, 10000] (* see A106856 *) 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 *) PROG (PARI) isok(p) = isprime(p) && !(kronecker(-23, p)==-1) && !polisirreducible(Mod(1, p)*(x^3-x-1)); \\ Arkadiusz Wesolowski, Oct 03 2021 (PARI) isok(p) = p==23 || (isprime(p) && #polrootsmod(x^3-x-1, p)==3); \\ Arkadiusz Wesolowski, Oct 09 2021 CROSSREFS Cf. A000594, A191021, A256567. Primes in A028958. Sequence in context: A274381 A044125 A044506 * A142107 A107208 A289735 Adjacent sequences:  A033214 A033215 A033216 * A033218 A033219 A033220 KEYWORD nonn,nice AUTHOR STATUS approved

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Last modified August 16 07:50 EDT 2022. Contains 356160 sequences. (Running on oeis4.)