A055506 Number of solutions to the equation phi(x) = n!.

2, 3, 4, 10, 17, 49, 93, 359, 1138, 3802, 12124, 52844, 182752, 696647, 2852886, 16423633, 75301815, 367900714, 1531612895, 8389371542, 40423852287, 213232272280, 1295095864798, 7991762413764, 42259876674716, 252869570952706, 1378634826630301, 8749244047999717

Offset: 1

Author

Labos Elemer, Jun 29 2000

Keywords

• nonn

Comments

Note that if phi(x) = n!, then x must be a product of primes p such that p - 1 divides n!. - David Wasserman, Apr 30 2002

Gives the row lengths of the table A165773 (see example). All solutions to phi(x)=n! are in the interval [n!,(n+1)!] with the smallest/largest solutions given in A055487/A165774 respectively. - M. F. Hasler, Oct 04 2009

Examples

			n = 5, phi(x) = 5! = 120 holds for the following 17 numbers: { 143, 155, 175, 183, 225, 231, 244, 248, 286, 308, 310, 350, 366, 372, 396, 450, 462 }.
From _M. F. Hasler_, Oct 04 2009: (Start)
The table A165773 looks as follows:
  1,2, (a(1)=2 numbers for which phi(n) = 1! = 1)
  3,4,6, (a(2)=3 numbers for which phi(n) = 2! = 2)
  7,9,14,18, (a(3)=4 numbers for which phi(n) = 3! = 6)
  35,39,45,52,56,70,72,78,84,90, (a(4)=10 numbers for which phi(n) = 4! = 24)
  ... (End)
		

Links

• Andrew Lelechenko, Table of n, a(n) for n = 1..36
• Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A000142, A000203, A014197, A054873, A055486, A067847, A165773, A165774.

Programs

PARI

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

Perl

use ntheory ":all"; print "$ ",scalar(inverse_totient(factorial($))),"\n" for 1..20; # Dana Jacobsen, Mar 04 2019

Formula

a(n) = A014197(n!) = Cardinality({x; A000010(x) = A000142(n)}).

Extensions

More terms from Jud McCranie, Jan 02 2001

More terms from David Wasserman, Apr 30 2002 (with the assistance of Vladeta Jovovic and Sascha Kurz)

a(21)-a(28) from Max Alekseyev, Jan 26 2012, Jul 09 2014