OFFSET
1,1
COMMENTS
Conjecture: (i) Any integer n > 1 can be written as k + m with k > 0 and m > 0 such that 2^k*(2^{phi(m)} - 1) + 1 is prime.
(ii) Each integer n > 2 can be written as k + m with k > 0 and m > 0 such that 2^k*(2^{phi(m)} - 1) - 1 is prime.
Part (i) of the conjecture implies that this sequence has infinitely many terms. See also A234399.
Note that the sequence contains all Fermat primes and Mersenne primes since 2^k + 1 = 2^k*(2^{phi(1)} - 1) + 1 and 2^p - 1 = 2*(2^{phi(p)} - 1) + 1, where k is a positive integer and p is a prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
EXAMPLE
a(1) = 3 since 2*(2^{phi(1)} - 1) + 1 = 3 is prime.
a(2) = 5 since 2^2*(2^{phi(1)} - 1) + 1 = 5 is prime.
a(3) = 7 since 2*(2^{phi(3)} - 1) + 1 = 7 is prime.
MATHEMATICA
S:=Intersection[Union[Table[EulerPhi[k], {k, 1, 5000}]], Table[k, {k, 1, 500}]]
n=0; Do[If[MemberQ[S, k]&&PrimeQ[2^m-2^(m-k)+1], n=n+1; Print[n, " ", 2^m-2^(m-k)+1]], {m, 1, 500}, {k, 1, m-1}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 25 2013
STATUS
approved