A058321 Number of x such that phi(x) = 2^n.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 0

Author

Labos Elemer, Dec 11 2000

Keywords

• nonn

Comments

If there are only 5 Fermat primes (A019434), then a(n) = 32 for n > 31. - T. D. Noe, Jun 21 2012 [Corrected by Jeppe Stig Nielsen, Oct 02 2021.]

The first unknown term is a(8589934592) which depends on whether A000215(33) is composite or prime. - Jeppe Stig Nielsen, Oct 02 2021

Examples

			For n = 0, a(0) = 2 because phi(1) = phi(2) = 1.
For n = 5, invphi(32) gives 7 values as follows: phi({51,64,68,80,96,102,120}) = {32,32,32,32,32,32,32}.
		

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 0..1000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• R. D. Carmichael, On Euler's phi-function, Bull. Amer. Math. Soc. 13 (1907), 241-243.
• R. D. Carmichael, Erratum: On Euler's phi-function, Bull. Amer. Math. Soc. 54 (1948), 1192.
• R. D. Carmichael, Erratum: Erratum: On Euler's phi-function, Bull. Amer. Math. Soc. 55 (1949), 212.
• Mathematics Stack Exchange, Empirical Observation on number of solutions to phi(n) = m.
• Eric W. Weisstein, MathWorld: Fermat prime.
• Wikipedia, Euler's totient function.

Crossrefs

Cf. A000010, A000079, A003401, A004729, A014197, A019434, A045544, A058213.

Programs

Maple

with(numtheory):[seq(nops(invphi(2^i)),i=1..100)];

PARI

a(n) = invphiNum(1 << n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Formula

a(n) = A014197(2^n) = A014197(A000079(n)).

Extensions

Added a(0) and corrected a(31) - T. D. Noe, Jun 21 2012

Correction of a(31) reverted; true value is a(31) = 33. - Jeppe Stig Nielsen, Oct 02 2021