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

Author

Hugo Pfoertner, Jun 20 2004

Keywords

• nonn

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

Examples

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

Links

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

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.

Programs

Magma

[n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015

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 *)

PARI

for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015

Formula

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

Extensions

Corrected by Zak Seidov, Mar 08 2006