|
%I #20 Sep 08 2022 08:45:53
%S 13,229,1013,1741,39317,64013,74101,157477,438989,551381,830597,
%T 1906637,2000389,4096013,7077901,9261013,10941061,15625013,16003021,
%U 21024589,24897101,27000013,69934541,74088013,79507013,93576677,122023949
%N Primes of the form k^3 + 13, k >= 0.
%C Necessarily, k = 6 * j or k = 6 * j + 4.
%C Values of k corresponding to terms of the sequence: 0, 6, 10, 12, 34, 40, 42, 54, 76, 82, 94, 124, 126, 160, 192, 210, 222, 250, 252, 276, 292, 300, 412, 420, 430, 454, 496, 502, 570, 586, 612, 622, 640, 670, 684, 712, 720, 724, 726, 756, 784, 822, 826, 874, 882, 894, 934, 952, 964, 1006, 1056.
%D H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag Berlin, 1973.
%H Robert Israel, <a href="/A176722/b176722.txt">Table of n, a(n) for n = 1..10000</a>
%H D. R. Heath-Brown and B. Z. Moroz, <a href="https://doi.org/10.1112/plms/84.2.257">Primes Represented by Binary Cubic Forms</a>, Proceedings of the London Math. Soc. vol. 84(2), pp. 257-288, 2002; see <a href="https://core.ac.uk/download/pdf/1568176.pdf">also</a>.
%e 0^3 + 13 = 13 = prime(6) = a(1);
%e 6^3 + 13 = 229 = prime(50) = a(2);
%e 300^3 + 13 = 27000013 = prime(1683067) = a(22).
%p select(isprime,[seq(seq((6*j+m)^3+13,m=[0,4]),j=0..1000)]); # _Robert Israel_, Jun 28 2018
%t Select[Range[0,1000]^3+13,PrimeQ] (* _Harvey P. Dale_, Mar 12 2011 *)
%o (Magma) [a: n in [0..500]|IsPrime(a) where a is n^3+13] // _Vincenzo Librandi_, Dec 22 2010
%Y Cf. A000040, A000578, A037396, A138375, A144953, A154648, A157772, A168147, A168219, A168327.
%K nonn
%O 1,1
%A Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 25 2010
|
