A050475 Numbers k such that x = 2^k-2 satisfies phi(x)+2 = phi(x+2).

3, 4, 6, 8, 14, 18, 20, 32, 62, 90, 108, 128, 522, 608, 1280, 2204, 2282, 3218, 4254, 4424, 9690, 9942, 11214, 19938, 21702, 23210, 44498, 86244, 110504, 132050, 216092, 756840, 859434, 1257788, 1398270, 2976222, 3021378, 6972594, 13466918, 20996012, 24036584, 25964952, 30402458, 32582658

Offset: 1

Author

Jud McCranie, Dec 24 1999

Keywords

• nonn

Comments

Other solutions of this equation are in A001838.

Also, numbers k such that 2^(k-1)-1 is prime. Proof: If x = 2^k-2, phi(x)+2 = phi(x+2) <==> phi(2^k-2)+2 = phi(2^k) <==> phi(2(2^(k-1)-1)) + 2 = 2^k(1-1/2) <==> phi(2)*phi(2^(k-1)-1)+2 = 2^(k-1) <==> phi(2^(k-1)-1) = 2^(k-1)-2 if y = 2^(k-1)-1. We have phi(y) = y-1 <==> y=2^(k-1)-1 is prime. Therefore a(n) = A000043(n)+1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004

Examples

			phi(2^18-2)+2 = 131072 = phi(2^18), so 18 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)

Crossrefs

Cf. A000043, A001838.

Programs

Mathematica

Flatten[Position[EulerPhi[2^# - 2] + 2 == EulerPhi[2^# ] & /@ Range[1, 250], True]] (* Vit Planocka *)

Formula

a(n) = A000043(n) + 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004

Extensions

a(39)-a(44) from Ivan Panchenko, Apr 11 2018