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

1,1

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

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.

Formula

Numbers k such that A068212(k) = 2.

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A242963 A191846 A068213 * A096989 A031950 A043694

Adjacent sequences: A065963 A065964 A065965 * A065967 A065968 A065969

Keyword

nonn

Author

Joseph L. Pe, Dec 08 2001

Extensions

More terms from Jason Earls, Dec 09 2001

Edited by Charles R Greathouse IV, Mar 18 2010

Status

approved