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 A237040 Semiprimes of the form k^3 + 1. 14

%I

%S 9,65,217,4097,5833,10649,21953,74089,195113,216001,343001,373249,

%T 474553,1000001,1061209,1191017,1404929,3241793,3796417,4251529,

%U 6859001,9261001,12487169,21952001,29791001,35937001,43614209,45882713,55742969,62099137,89915393,94818817,117649001

%N Semiprimes of the form k^3 + 1.

%C k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.

%C Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?

%C In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - _Charles R Greathouse IV_, Jul 02 2017

%H Vincenzo Librandi, <a href="/A237040/b237040.txt">Table of n, a(n) for n = 1..1400</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Semiprime">Semiprime</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Landau&#39;s_problems">Landau's problems</a>

%F a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

%e 9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

%t L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1

%t Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* _Zak Seidov_, Jun 26 2017 *)

%o (PARI) lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ _Michel Marcus_, Jun 27 2017

%o (PARI) list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ _Charles R Greathouse IV_, Jul 02 2017

%o (MAGMA) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // _Vincenzo Librandi_, Jul 02 2017

%Y Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.

%Y Cf. A242262 (semiprimes of the form k^3 - 1).

%K nonn

%O 1,1

%A _Jonathan Sondow_, Feb 02 2014

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Last modified August 16 00:25 EDT 2018. Contains 313782 sequences. (Running on oeis4.)