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Primes p such that (2*p)^3 + 1 is a semiprime.
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%I #20 Jun 28 2021 13:16:31

%S 2,3,11,29,53,179,191,491,641,659,683,1103,1499,1901,2129,2543,2549,

%T 3803,3851,4271,4733,4943,5303,5441,6101,6329,6449,7193,7211,8093,

%U 8513,9059,9419,10091,10271,10733,10781,11321,12203,12821,13451,14561,15233,15803,17159,17333,18131,19373,19919

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

%C 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)).

%C Primes in A237037.

%C 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

%H Amiram Eldar, <a href="/A237038/b237038.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain prime</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Semiprime">Semiprime</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>.

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

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

%t Select[Range[20000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]

%t Select[Prime[Range[2500]],PrimeOmega[(2#)^3+1]==2&] (* _Harvey P. Dale_, Jun 28 2021 *)

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

%K nonn

%O 1,1

%A _Jonathan Sondow_, Feb 02 2014