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

1,1

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

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).

Formula

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

Example

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)

Prog

(Perl) use ntheory ":all"; print "$_ ", scalar(inverse_totient(factorial($_))), "\n" for 1..20; # Dana Jacobsen, Mar 04 2019
(PARI) a(n) = invphiNum(n!); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Crossrefs

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

Sequence in context: A056701 A365352 A285190 * A329660 A098088 A080500

Adjacent sequences: A055503 A055504 A055505 * A055507 A055508 A055509

Keyword

nonn

Author

Labos Elemer, Jun 29 2000

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

Status

approved