A070162 Numbers k such that k - phi(k) - 1 is a prime.

6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 26, 34, 36, 38, 40, 42, 44, 46, 48, 50, 56, 58, 60, 62, 64, 72, 74, 78, 80, 82, 84, 86, 88, 92, 94, 100, 106, 108, 116, 118, 122, 126, 134, 136, 142, 146, 150, 152, 156, 158, 162, 164, 166, 178, 180, 182, 192, 194, 198, 202, 204

Offset: 1

Author

Labos Elemer, Apr 26 2002

Keywords

• nonn

Comments

Maximal solution is either k = 2p or, in the case of Mersenne primes M, k = 2(M+1) and f(k) = p or f(k) = M.

Examples

			k = 192: phi(192) = 64, cototient(192) = 128, k - phi(192) - 1 = 127 is prime.
k = 2p: 2p - phi(2p) - 1 = 2p - p + 1 - 1 = p, so 2*prime is always a solution.
k = 2^(q+1), where q is a Mersenne prime exponent: cototient(k) - 1 = 2^(p+1) - 2^p - 1 = 2^p - 1, which is the corresponding Mersenne prime.
Numbers k in {192, 224, 248, 254, 256} give p = 127.
numbers k in {72, 80, 88, 92, 94} give p = 47.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A051953.

Programs

Mathematica

Do[s=n-EulerPhi[n]-1; If[PrimeQ[s], Print[n, s]], n, 1, 10000]
Select[Range[250],PrimeQ[#-EulerPhi[#]-1]&] (* Harvey P. Dale, Sep 05 2023 *)

PARI

is(k) = isprime(k - eulerphi(k) - 1); \\ Amiram Eldar, Nov 07 2024

Formula

f(k) = k - A000010(k) - 1 = A051953(k) - 1 is prime.