

A067319


Numbers n such that phi(n)^phi(n)+1 is prime.


0




OFFSET

1,2


COMMENTS

It is conjectured that the sequence of Fermat primes (A019434) is complete; if so then this sequence is complete:
Suppose that x is a positive integer for which x^x+1 is prime. If x has an odd prime factor p, then x^x + 1 = (x^(x/p))^p + 1 is divisible by x^(x/p) + 1, so it is not prime. So x must be a power of 2. Hence x^x is also a power of 2, so x^x+1 is a Fermat prime.
If there are no Fermat primes beyond the known ones (as is widely believed), then x must be 1, 2, or 4. Letting x=phi(n), it is easy to see that n must be one of the numbers listed.  Dean Hickerson, Feb 11 2002


LINKS

Table of n, a(n) for n=1..9.


EXAMPLE

Cases n=112 are based on the primes 2, 5, 257.


MATHEMATICA

ephiQ[n_]:=Module[{eph=EulerPhi[n]}, PrimeQ[eph^eph+1]]; Select[ Range[ 20], ephiQ] (* Harvey P. Dale, Feb 23 2021 *)


PROG

(PARI) isok(n) = isprime(eulerphi(n)^eulerphi(n) + 1); \\ Michel Marcus, Oct 07 2013


CROSSREFS

Cf. A063439, A000010.
Sequence in context: A253012 A036409 A005423 * A086049 A328208 A173643
Adjacent sequences: A067316 A067317 A067318 * A067320 A067321 A067322


KEYWORD

nonn


AUTHOR

Labos Elemer, Jan 15 2002


STATUS

approved



