A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.

1, 2, 3, 8, 9, 11, 14, 21, 29, 30, 35, 36, 39, 50, 51, 53, 56, 74, 78, 81, 95, 105, 116, 140, 155, 165, 176, 179, 191, 198, 224, 228, 245, 284, 303, 336, 378, 393, 410, 413, 414, 428, 429, 438, 464, 485, 491, 504, 506, 515, 534, 546, 554, 575, 596, 611, 638, 641, 648, 659, 680, 683, 711, 714, 725, 744, 765, 774, 791

Offset: 1

Author

Jonathan Sondow, Feb 02 2014

Keywords

• nonn

Comments

Numbers k such that 2*k+1 and 4*k^2 - 2*k + 1 are both prime.

Same as k/2 such that k^3 + 1 is a semiprime, because then k must be even.

Examples

			(2*1)^3 + 1 = 9 = 3*3 is a semiprime, so a(1) = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Semiprime.
• Wikipedia, Semiprime.

Crossrefs

Cf. A001358, A046315, A081256, A096173, A096174, A237038, A237039, A237040.

Programs

Mathematica

Select[Range[800], PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Range[800],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Nov 28 2024 *)

Formula

a(n) = A096173(n)/2 = (1/2)*(A237040(n)-1)^(1/3).