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A176584 Primes of the form p = k^3 + (largest square <= k^3).

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%I A176584 #14 Jan 11 2024 00:56:44
%S A176584 2,194057,601067,1607173,3283993,3630257,3721363,4918999,5255713,
%T A176584 5848307,6749569,9999907,17992913,20432729,23393483,34746203,44845993,
%U A176584 73843813,84277273,107165033,109735649,120469409,125138873,130898927
%N A176584 Primes of the form p = k^3 + (largest square <= k^3).
%C A176584 Consider k = 4 * x^2 + 1, where the largest square <= k^3 is (8 * x^3 + 3 * x)^2.  Bunyakowsky's conjecture implies there are infinitely many primes of the form (4 * x^2 + 1)^3 + (8 * x^3 + 3 * x)^2, and so that the sequence is infinite.  The first term of this form is 93389778901 = a(165), corresponding to x = 30. - _Robert Israel_, Jan 10 2024
%H A176584 Robert Israel, <a href="/A176584/b176584.txt">Table of n, a(n) for n = 1..10000</a>
%p A176584 g:= n -> n + floor(sqrt(n))^2:
%p A176584 select(isprime,map(g,[seq(i^3,i=1..1000)])); # _Robert Israel_, Jan 10 2024
%t A176584 r[n_]:=n^3;f[n_]:=r[n]+Floor[Sqrt[r[n]]]^2;Select[Table[f[n],{n,0,6!}],PrimeQ[ # ]&]
%o A176584 (PARI) select(isprime, vector(1000, n, n^3+sqrtint(n^3)^2)) \\ _Michel Marcus_, Jan 10 2024
%Y A176584 Primes in A176580.
%Y A176584 Cf. A065733, A075847, A077116, A176581, A176582, A176583.
%K A176584 nonn
%O A176584 1,1
%A A176584 _Vladimir Joseph Stephan Orlovsky_, Apr 21 2010