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A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.

%I A127936 #30 Jun 07 2025 23:55:57
%S A127936 1,2,3,5,6,8,9,11,15,21,30,39,50,63,83,95,99,156,173,350,854,1308,
%T A127936 1769,2903,5250,5345,5639,6195,7239,21368,41669,47684,58619,63515,
%U A127936 69468,70539,133508,134993,187160,493095
%N A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.
%C A127936 If this sequence is infinite then so is A124401.
%C A127936 Equals A127965(n)/2.
%C A127936 The sum has the simple closed form 1 + 2/3*(4^n-1). - _Stefan Steinerberger_, Nov 24 2007
%C A127936 Terms beyond a(30) correspond to probable primes, cf. A000978. - _M. F. Hasler_, Aug 29 2008
%F A127936 a(n) = floor(A000978(n)/2) = ceiling(log(4)(A000979(n))); A000978(n) = 2 a(n) + 1; A000979(n) = (2*4^a(n)+1)/3. - _M. F. Hasler_, Aug 29 2008
%e A127936 a(1)=1 because 1 + 2 = 3 is prime;
%e A127936 a(2)=2 because 1 + 2 + 2^3 = 11 is prime;
%e A127936 a(3)=3 because 1 + 2 + 2^3 + 2^5 = 43 is prime;
%e A127936 a(4)=5 because 1 + 2 + 2^3 + 2^5 + 2^7 + 2^9 = 683 is prime;
%e A127936 ...
%t A127936 a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]], AppendTo[a, x]], {x, 1, 1000}]; a
%t A127936 b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, (1/2)(DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1])], {x, 1, Length[a]}]; d
%t A127936 Position[Accumulate[2^(2*Range[1000]-1)],_?(PrimeQ[#+1]&)]//Flatten (* The program generates the first 21 terms of the sequence. To generate more, increase the Range constant. *) (* _Harvey P. Dale_, Mar 23 2022 *)
%o A127936 (PARI) for(n=1,999, ispseudoprime(2^(2*n+1)\3+1) & print1(n",")) \\ _M. F. Hasler_, Aug 29 2008
%o A127936 (Haskell)
%o A127936 import Data.List (findIndices)
%o A127936 a127936 n = a127936_list !! (n-1)
%o A127936 a127936_list = findIndices ((== 1) . a010051'') a007583_list
%o A127936 -- _Reinhard Zumkeller_, Mar 24 2013
%o A127936 (Python)
%o A127936 from sympy import isprime
%o A127936 A127936 = [i for i in range(1,10**3) if isprime(int('01'*i+'1',2))]
%o A127936 # _Chai Wah Wu_, Sep 05 2014
%Y A127936 Cf. A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936, A127936, A124401, A010051, A007583.
%K A127936 nonn,more
%O A127936 1,2
%A A127936 _Artur Jasinski_, Feb 08 2007, Feb 09 2007
%E A127936 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 11 2007
%E A127936 2 more terms from _Stefan Steinerberger_, Nov 24 2007
%E A127936 6 more terms from _Dmitry Kamenetsky_, Jul 12 2008
%E A127936 a(30)-a(40) calculated from A000978 by _M. F. Hasler_, Aug 29 2008