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

%I A237037 #14 Feb 16 2025 08:33:21
%S A237037 1,2,3,8,9,11,14,21,29,30,35,36,39,50,51,53,56,74,78,81,95,105,116,
%T A237037 140,155,165,176,179,191,198,224,228,245,284,303,336,378,393,410,413,
%U A237037 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
%N A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.
%C A237037 Numbers k such that 2*k+1 and 4*k^2 - 2*k + 1 are both prime.
%C A237037 Same as k/2 such that k^3 + 1 is a semiprime, because then k must be even.
%H A237037 Amiram Eldar, <a href="/A237037/b237037.txt">Table of n, a(n) for n = 1..10000</a>
%H A237037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.
%H A237037 Wikipedia, <a href="https://en.wikipedia.org/wiki/Semiprime">Semiprime</a>.
%F A237037 a(n) = A096173(n)/2 = (1/2)*(A237040(n)-1)^(1/3).
%e A237037 (2*1)^3 + 1 = 9 = 3*3 is a semiprime, so a(1) = 1.
%t A237037 Select[Range[800], PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
%t A237037 Select[Range[800],PrimeOmega[(2#)^3+1]==2&] (* _Harvey P. Dale_, Nov 28 2024 *)
%Y A237037 Cf. A001358, A046315, A081256, A096173, A096174, A237038, A237039, A237040.
%K A237037 nonn
%O A237037 1,2
%A A237037 _Jonathan Sondow_, Feb 02 2014