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

1, 2, 3, 5, 6, 8, 9, 11, 15, 21, 30, 39, 50, 63, 83, 95, 99, 156, 173, 350, 854, 1308, 1769, 2903, 5250, 5345, 5639, 6195, 7239, 21368, 41669, 47684, 58619, 63515, 69468, 70539, 133508, 134993, 187160, 493095

Offset

1,2

Comments

If this sequence is infinite then so is A124401.

Equals A127965(n)/2.

The sum has the simple closed form 1 + 2/3*(4^n-1). - Stefan Steinerberger, Nov 24 2007

Terms beyond a(30) correspond to probable primes, cf. A000978. - M. F. Hasler, Aug 29 2008

Links

Table of n, a(n) for n=1..40.

Formula

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

Example

a(1)=1 because 1 + 2 = 3 is prime;
a(2)=2 because 1 + 2 + 2^3 = 11 is prime;
a(3)=3 because 1 + 2 + 2^3 + 2^5 = 43 is prime;
a(4)=5 because 1 + 2 + 2^3 + 2^5 + 2^7 + 2^9 = 683 is prime;
...

Mathematica

a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]], AppendTo[a, x]], {x, 1, 1000}]; a
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
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 *)

Prog

(PARI) for(n=1, 999, ispseudoprime(2^(2*n+1)\3+1) & print1(n", ")) \\ M. F. Hasler, Aug 29 2008
(Haskell)
import Data.List (findIndices)
a127936 n = a127936_list !! (n-1)
a127936_list = findIndices ((== 1) . a010051'') a007583_list
-- Reinhard Zumkeller, Mar 24 2013
(Python)
from sympy import isprime
A127936 = [i for i in range(1, 10**3) if isprime(int('01'*i+'1', 2))]
# Chai Wah Wu, Sep 05 2014

Crossrefs

Cf. A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936, A127936, A124401, A010051, A007583.

Sequence in context: A335659 A000534 A136112 * A280771 A368363 A348195

Adjacent sequences: A127933 A127934 A127935 * A127937 A127938 A127939

Keyword

nonn,more

Author

Artur Jasinski, Feb 08 2007, Feb 09 2007

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 11 2007

2 more terms from Stefan Steinerberger, Nov 24 2007

6 more terms from Dmitry Kamenetsky, Jul 12 2008

a(30)-a(40) calculated from A000978 by M. F. Hasler, Aug 29 2008

Status

approved