A065966 Numbers k such that phi(k) / 2 is prime.

5, 7, 8, 9, 10, 11, 12, 14, 18, 22, 23, 46, 47, 59, 83, 94, 107, 118, 166, 167, 179, 214, 227, 263, 334, 347, 358, 359, 383, 454, 467, 479, 503, 526, 563, 587, 694, 718, 719, 766, 839, 863, 887, 934, 958, 983, 1006, 1019, 1126, 1174, 1187, 1283, 1307, 1319

Offset: 1

Author

Joseph L. Pe, Dec 08 2001

Keywords

• nonn

Comments

This is probably an infinite sequence, but a proof would be nice. Are there infinitely many consecutive terms of the sequence which are also consecutive integers? (For example, 7, 8 and 46, 47.)

Apart from 8, 9, 12 and 18, all the terms of the sequence are safe primes or twice safe primes. It is not known if there are infinitely many safe primes (cf. A005385, A005384). For consecutive terms of the sequence which are also consecutive integers see A066179. - Vladeta Jovovic, Dec 16 2001

Examples

			phi(46)/2 = 22/2 = 11, a prime.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• André Hernández-Espiet, Reyes M. Ortiz-Albino, On the Characterization of tau(n)-Atoms, arXiv:1905.02834 [math.NT], 2019. See Proposition 3.1.

Crossrefs

Cf. A000010, A005385, A005384, A066179, A006530, A052126, A068211, A068212.

Programs

Mathematica

Select[Range[1400],PrimeQ[EulerPhi[#]/2]&] (* Harvey P. Dale, Feb 11 2020 *)

PARI

for(n=3,5000, if(isprime(eulerphi(n)/2),print1(n,",")))

PARI

{ n=0; for (m=3, 10^9, if (isprime(eulerphi(m)/2), write("b065966.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 05 2009

Formula

Numbers k such that A068212(k) = 2.

Extensions

More terms from Jason Earls, Dec 09 2001

Edited by Charles R Greathouse IV, Mar 18 2010