A237183 Primes p with phi(p+1) - 1 and phi(p+1) + 1 both prime, where phi(.) is Euler's totient function.

7, 11, 13, 17, 37, 41, 53, 61, 97, 151, 181, 197, 227, 233, 251, 269, 277, 397, 433, 457, 487, 541, 557, 571, 593, 619, 631, 719, 743, 769, 839, 857, 929, 941, 947, 953, 1013, 1021, 1049, 1061, 1063, 1201, 1237, 1277, 1307, 1321, 1367, 1481, 1511, 1549

Offset: 1

Author

Zhi-Wei Sun, Feb 04 2014

Keywords

• nonn

Comments

According to part (i) of the conjecture in A237168, this sequence should have infinitely many terms.

Examples

			a(1) = 7 since 7, phi(7+1) - 1 = 3 and phi(7+1) + 1 = 5 are all prime, but phi(2+1) - 1 = phi(3+1) - 1 = phi(5+1) - 1 = 1 is not prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000040, A001359, A006512, A072281, A237127, A237130, A237168.

Programs

Mathematica

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
n=0;Do[If[PQ[Prime[k]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
Select[Prime[Range[300]],And@@PrimeQ[EulerPhi[#+1]+{1,-1}]&] (* Harvey P. Dale, Mar 06 2014 *)

PARI

s=[]; forprime(p=2, 2000, if(isprime(eulerphi(p+1)-1) && isprime(eulerphi(p+1)+1), s=concat(s, p))); s \\ Colin Barker, Feb 04 2014