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A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

%I A242116 #34 Mar 18 2025 10:33:24
%S A242116 9,25,65,161,2049,4609,22529,1048577,44040193,283467841537,
%T A242116 1202590842881,256065421246102339102334047485953,
%U A242116 4259306016766850789028922770063361,356615920533143509709616588588493085605889,57729314674570665269045550892293179276409335447553
%N A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.
%C A242116 The k-th Cullen number Cullen(k) = k*2^k + 1.
%C A242116 If Cullen(k) is semiprime, it is in the sequence.
%C A242116 The next term, a(16), has 52 digits.
%H A242116 Tyler Busby, <a href="/A242116/b242116.txt">Table of n, a(n) for n = 1..35</a> (terms 1..16 from K. D. Bajpai, terms 17..34 from Amiram Eldar)
%F A242116 a(n) = A002064(A242175(n)). - _Amiram Eldar_, Nov 27 2019
%e A242116 a(4) = 161 = (5*2^5+1) is 5th Cullen number and 161 = 7 * 23 is semiprime.
%e A242116 a(5) = 2049 = (8*2^8+1) is 8th Cullen number and 2049 = 3 * 683 is semiprime.
%p A242116 with(numtheory): A242116:= proc(); if bigomega(x*2^x+1) = 2 then RETURN (x*2^x+1);  fi; end: seq(A242116 (), x=1..200);
%t A242116 cullen[n_] := n * 2^n + 1; Select[cullen[Range[35]], PrimeOmega[#] == 2 &] (* _Amiram Eldar_, Nov 27 2019 *)
%o A242116 (PARI) select(n->bigomega(n)==2, vector(90,n,n<<n+1)) \\ _Charles R Greathouse IV_, May 06 2014
%o A242116 (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [s: n in [1..200] | IsSemiprime(s) where s is n*2^n+1]; // // _Vincenzo Librandi_, May 07 2014
%Y A242116 Cf. A001358, A002064, A003261, A005849, A242175.
%K A242116 nonn
%O A242116 1,1
%A A242116 _K. D. Bajpai_, May 04 2014