A334993 - OEIS

%I #18 Feb 11 2021 10:19:40

%S 1,5,9,17,57,65,897,4217,6225,152529,3648969,5570081

%N Numbers k such that 2*3^k + 1 is prime and divides Phi(3^m, 2).

%C A subset of odd values from A003306.

%C If p = 2*3^k + 1 is prime then p divides 2^(3^k) + (-1)^k, due to Euler's criterion.

%C Only odd terms of sequence A003306 can divide the cyclotomic expression Phi(3^m, 2); none of the even terms of sequence A003306 can divide 2^3^k-1 and therefore cannot divide Phi(3^m, 2).

%H C. Caldwell's <a href="https://primes.utm.edu/top20/page.php?id=37">Prime Page for Divides Phi category</a>.

%o (PARI) dp(n)=Mod(2,2*3^n+1)^3^n==1;

%o forstep(n=1,6225,2,if(dp(n),print1(n,", ")))

%Y Cf. A003306.

%K nonn,hard,more

%O 1,2

%A _Serge Batalov_, May 18 2020