

A055506


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


6



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
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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


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


CROSSREFS

Cf. A000142, A000010, A014197, A000203, A054873, A067847, A055486, A165774.
Sequence in context: A220402 A056701 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



