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Smallest number x such that cototient(x) = 2^n.
8

%I #43 Aug 13 2022 17:59:14

%S 2,4,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,

%T 196608,393216,786432,1572864,3145728,6291456,12582912,25165824,

%U 50331648,100663296,201326592,402653184,805306368,1610612736,3221225472

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

%C 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

%C 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

%H Jud McCranie, <a href="/A058764/b058764.txt">Table of n, a(n) for n = 0..46</a>

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

%F 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

%e 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.

%t 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 *)

%o (PARI) a(n) = {x = 1; while(x - eulerphi(x) != 2^n, x++); x;} \\ _Michel Marcus_, Dec 11 2013

%o (PARI) a(n) = if(n>1,3,4)<<(n-1) \\ _M. F. Hasler_, Nov 10 2016

%Y Cf. A051953, A053579, A053650.

%Y Cf. A042950. - _R. J. Mathar_, Jan 30 2009

%Y Cf. A007283.

%K nonn,hard

%O 0,1

%A _Labos Elemer_, Jan 02 2001

%E Edited by _M. F. Hasler_, Nov 10 2016

%E a(27)-a(31) from _Jud McCranie_, Jul 13 2017