A058215 Largest solution of phi(x) = 2^n.

2, 6, 12, 30, 60, 120, 240, 510, 1020, 2040, 4080, 8160, 16320, 32640, 65280, 131070, 262140, 524280, 1048560, 2097120, 4194240, 8388480, 16776960, 33553920, 67107840, 134215680, 268431360, 536862720, 1073725440, 2147450880, 4294901760, 8589934590

Offset: 0

Author

Labos Elemer, Nov 30 2000

Keywords

• nonn

Comments

The ratio of adjacent terms is 2 except for five terms (if there are exactly five Fermat primes). - T. D. Noe, Jun 21 2012

Examples

			For n = 6, 2^n = 64; the solutions of phi(x) = 64 are {85,128,136,160,170,192,204,240}; the largest is a(6) = 240.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A001317, A003401, A004729, A019434, A045544, A047999, A053576, A054432, A058213, A058214.

Programs

Mathematica

phiinv[ n_, pl_ ] := Module[ {i, p, e, pe, val}, If[ pl=={}, Return[ If[ n==1, {1}, {} ] ] ]; val={}; p=Last[ pl ]; For[ e=0; pe=1, e==0||Mod[ n, (p-1)pe/p ]==0, e++; pe*=p, val=Join[ val, pe*phiinv[ If[ e==0, n, n*p/pe/(p-1) ], Drop[ pl, -1 ] ] ] ]; Sort[ val ] ]; phiinv[ n_ ] := phiinv[ n, Select[ 1+Divisors[ n ], PrimeQ ] ]; Table[ phiinv[ 2^n ][ [ -1 ] ], {n, 0, 30} ] (* phiinv[ n, pl ] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[ n ] = list of x with phi(x)=n *)

PARI

a(n) = invphiMax(2^n); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Formula

Assuming there are only 5 Fermat primes (A019434), a(n) = 2^(n-30)*(2^32-1) for n >= 31.

Extensions

Edited by Dean Hickerson, Jan 25 2002