A058764 Smallest number x such that cototient(x) = 2^n.

2, 4, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Author

Labos Elemer, Jan 02 2001

Keywords

• nonn
• hard

Comments

Since the cototient of 3*2^n is 2^(n+1), upper bounds are given by A007283(n-1). - R. J. Mathar, Oct 13 2008

A058764(n+1) is the number of different walks with n steps in the graph G = ({1,2,3,4}, {{1,2}, {2,3}, {3,4}}). - Aldo González Lorenzo, Feb 27 2012

Examples

			a(5) = 48, cototient(48) = 48-Phi(48) = 48-16 = 32. For n>2, a(n) = 3*2^(n-1); largest solutions = 2^(n+1). Prime factors of solutions: 2 and Mersenne-primes were found only.
		

Links

• Jud McCranie, Table of n, a(n) for n = 0..46

Crossrefs

Cf. A051953, A053579, A053650.

Cf. A042950. - R. J. Mathar, Jan 30 2009

Cf. A007283.

Programs

Mathematica

Function[s, Flatten@ Map[First@ Position[s, #] &, 2^Range[0, Floor@ Log2@ Max@ s]]]@ Table[n - EulerPhi@ n, {n, 10^7}] (* Michael De Vlieger, Dec 17 2016 *)

PARI

a(n) = {x = 1; while(x - eulerphi(x) != 2^n, x++); x;} \\ Michel Marcus, Dec 11 2013

PARI

a(n) = if(n>1,3,4)<<(n-1) \\ M. F. Hasler, Nov 10 2016

Formula

a(n) = min { x | A051953(x) = 2^n }.

a(n) = (if n>1 then 3 else 4)*2^(n-1) = A007283(n-1) for n>1. (Conjectured.) - M. F. Hasler, Nov 10 2016

Extensions

Edited by M. F. Hasler, Nov 10 2016

a(27)-a(31) from Jud McCranie, Jul 13 2017