login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime. 13

%I

%S 2,2,21,209,72,260,17,3311,4469,94259

%N Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

%C Numbers of the form k^n*(k^n+1) + 1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even k^(4*n) + k^(2*n) + 1 = (k^(2*n)+1)^2 - (k^n)^2 = (k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3 is k^n*(k^n+1) + 1 always composite?

%C Phi[3^(n+1),k] = k^(3^n)*(k^(3^n)+1)+1. When m <> 3^n in k^m*(k^m+1)+1, Phi[3m,k] < k^m*(k^m+1)+1 and is a divisor of it. - _Lei Zhou_, Feb 09 2012

%C The prime number corresponding to the 10th term is a 587458-digit number. - _Lei Zhou_, Jul 04 2014

%H Lei Zhou, <a href="http://primes.utm.edu/primes/page.php?id=118125">Prime Database Entry</a>, July 4th, 2014.

%t Table[i = 1; m = 3^u; While[i++; cp = 1 + i^m + i^(2*m); ! PrimeQ[cp]]; i, {u, 1, 7}] (* _Lei Zhou_, Feb 01 2012 *)

%Y Cf. A101406, A153436, A056993.

%K nonn,more,hard

%O 1,1

%A _Pierre CAMI_, Dec 26 2008

%E 3311 from _Lei Zhou_ using OpenPFGW, Feb 01 2012

%E 4469 from _Lei Zhou_ using OpenPFGW, Feb 09 2012

%E New term, 94259, from _Lei Zhou_ using OpenPFGW, Jul 04 2014

%E Name and Comment corrected by _Robert Price_, Nov 11 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 7 10:14 EST 2021. Contains 341869 sequences. (Running on oeis4.)