A237040 Semiprimes of the form k^3 + 1.

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001

Offset: 1

Author

Jonathan Sondow, Feb 02 2014

Keywords

• nonn

Comments

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

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?

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1400
• Eric Weisstein's World of Mathematics, Semiprime
• Wikipedia, Semiprime
• Wikipedia, Landau's problems

Crossrefs

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

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

Programs

Magma

IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017

Mathematica

L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)

PARI

lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017

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

Formula

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