login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A237183
Primes p with phi(p+1) - 1 and phi(p+1) + 1 both prime, where phi(.) is Euler's totient function.
2
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
1,1
COMMENTS
According to part (i) of the conjecture in A237168, this sequence should have infinitely many terms.
EXAMPLE
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.
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 *)
PROG
(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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2014
STATUS
approved