

A059917


a(n) = (3^(2^n) + 1)/2 = A059919(n)/2, n >= 0.


5




OFFSET

0,1


COMMENTS

Average of first 2^(n+1) powers of 3 divided by average of first 2^n powers of 3.
Numerator of b(n) where b(n) = (1/2)*(b(n1) + 1/b(n1)), b(0)=2.  Vladeta Jovovic, Aug 15 2002
From Daniel Forgues, Jun 22 2011: (Start)
Since for the generalized Fermat numbers 3^(2^n)+1 (A059919), we have a(n) = 2*a(n1)*a(n2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 2*(empty product, i.e., 1) + 2 = 4 = a(0). This formula implies that the GCD of any pair of terms of A059919 is 2, which means that the terms of (3^(2^n)+1)/2 (A059917) are pairwise coprime.
2, 5, 41, 21523361, 926510094425921 are prime. 3281 = 17*193. (End)
a(0), a(1), a(2), a(4), a(5), and a(6) are prime. Conjecture: a(n) is composite for all n > 6.  Thomas Ordowski, Dec 26 2012
This may be a primality test for Mersenne numbers. a(2) = 41 == 1 mod 7 (=M3), a(4) = 21523361 == 30 == 1 mod 31 (=M5). However, a(10) is not == 1 mod M11.  Nobuyuki Fujita, May 16 2015


LINKS

Harry J. Smith, Table of n, a(n) for n = 0..11
A. Granville, Using Dynamical Systems to Construct Infinitely Many Primes, arXiv:1708.06953 [math.NT], 2017.
A. Granville, Using Dynamical Systems to Construct Infinitely Many Primes, The American Mathematical Monthly 125, no. 6 (2018), 483496. DOI: 10.1080/00029890.2018.1447732


FORMULA

a(n) = a(n1)*(3^(2^(n1)) + 1)  3^(2^(n1)) = A059723(n+1)/A059723(n) = A059918(n) + 1 = a(n1)*A059919(n1)  A011764(n1).
a(0) = 2; a(n) = ((2*a(n1)  1)^2 + 1)/2, n >= 1.  Daniel Forgues, Jun 22 2011


EXAMPLE

a(2) = Average(1,3,9,27,81,243,729,2187)/Average(1,3,9,27) = 410/10 = 41.


MAPLE

seq((3^(2^n)+1)/2, n=0..11); # Muniru A Asiru, Aug 07 2018


MATHEMATICA

Table[(3^(2^n) + 1)/2, {n, 0, 10}] (* Vincenzo Librandi, May 16 2015 *)


PROG

(PARI) { for (n=0, 11, write("b059917.txt", n, " ", (3^(2^n) + 1)/2); ) } \\ Harry J. Smith, Jun 30 2009
(Magma) [(3^(2^n)+1)/2: n in [0..10]]; // Vincenzo Librandi, May 16 2015
(GAP) List([0..10], n>(3^(2^n)+1)/2); # Muniru A Asiru, Aug 07 2018


CROSSREFS

Cf. A059918, A059919. Primes are in A093625.
Sequence in context: A126469 A054859 A076725 * A255963 A093625 A042447
Adjacent sequences: A059914 A059915 A059916 * A059918 A059919 A059920


KEYWORD

nonn,frac


AUTHOR

Henry Bottomley, Feb 08 2001


STATUS

approved



