A237038 Primes p such that (2*p)^3 + 1 is a semiprime.

2, 3, 11, 29, 53, 179, 191, 491, 641, 659, 683, 1103, 1499, 1901, 2129, 2543, 2549, 3803, 3851, 4271, 4733, 4943, 5303, 5441, 6101, 6329, 6449, 7193, 7211, 8093, 8513, 9059, 9419, 10091, 10271, 10733, 10781, 11321, 12203, 12821, 13451, 14561, 15233, 15803, 17159, 17333, 18131, 19373, 19919

Offset: 1

Author

Jonathan Sondow, Feb 02 2014

Keywords

• nonn

Comments

Same as Sophie Germain primes p such that 4*p^2 - 2*p + 1 is also prime (because (2*p)^3 + 1 = (2*p + 1)(4*p^2 - 2*p + 1)).

Primes in A237037.

For n>1, 8*a(n)^3 is a solution for the equation phi(x+1) - phi(x) = x/2. - Farideh Firoozbakht, Dec 17 2014

Examples

			11 is prime and (2*11)^3 + 1 = 10649 = 23*463 is a semiprime, so 11 is a member.
		

Links

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

Crossrefs

Cf. A000010, A001358, A005384, A046315, A081256, A096173, A096174, A237037, A237039, A237040.

Programs

Mathematica

Select[Range[20000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Prime[Range[2500]],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Jun 28 2021 *)

Formula

a(n) = (1/2)*(A237039(n)-1)^(1/3).