A096173 Numbers k such that k^3+1 is an odd semiprime.

2, 4, 6, 16, 18, 22, 28, 42, 58, 60, 70, 72, 78, 100, 102, 106, 112, 148, 156, 162, 190, 210, 232, 280, 310, 330, 352, 358, 382, 396, 448, 456, 490, 568, 606, 672, 756, 786, 820, 826, 828, 856, 858, 876, 928, 970, 982, 1008, 1012, 1030, 1068, 1092, 1108, 1150

Offset

1,1

Comments

Numbers n such that n^3 + 1 is a semiprime, because then n^3 + 1 must be odd, since n^3 + 1 = (n+1)*(n^2 - n + 1) is a semiprime only if n+1 is odd. - Jonathan Sondow, Feb 02 2014

Obviously, n + 1 is always a prime number. Sequence is intersection of A006093 and A055494. - Altug Alkan, Dec 20 2015

Links

R. J. Mathar, Table of n, a(n) for n = 1..2086

Formula

a(n) = 2*A237037(n) = (A237040(n)-1)^(1/3). - Jonathan Sondow, Feb 02 2014

Example

a(1)=2 because 2^3+1=9=3*3, a(13)=100: 100^3+1=1000001=101*9901.

Maple

select(n -> isprime(n+1) and isprime(n^2-n+1), [seq(2*i, i=1..1000)]); # Robert Israel, Dec 20 2015

Mathematica

Select[Range[1200], PrimeQ[#^2 - # + 1] && PrimeQ[# + 1] &] (* Jonathan Sondow, Feb 02 2014 *)

Prog

(PARI) for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015
(Magma) [n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015

Crossrefs

Cf. A001358; A081256: largest prime factor of k^3+1; A096174: (k^3+1)/(k+1) is prime; A046315, A237037, A237038, A237039, A237040.

Cf. A006093, A055494.

Sequence in context: A326782 A358126 A096174 * A287681 A333021 A114874

Adjacent sequences: A096170 A096171 A096172 * A096174 A096175 A096176

Keyword

nonn

Author

Hugo Pfoertner, Jun 20 2004

Extensions

Corrected by Zak Seidov, Mar 08 2006

Status

approved