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%I #46 Mar 04 2019 17:12:30
%S 2,3,4,10,17,49,93,359,1138,3802,12124,52844,182752,696647,2852886,
%T 16423633,75301815,367900714,1531612895,8389371542,40423852287,
%U 213232272280,1295095864798,7991762413764,42259876674716,252869570952706,1378634826630301,8749244047999717
%N Number of solutions to the equation phi(x) = n!.
%C 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
%C 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
%H Andrew Lelechenko, <a href="/A055506/b055506.txt">Table of n, a(n) for n = 1..36</a>
%H Max A. Alekseyev, <a href="https://www.emis.de/journals/JIS/VOL19/Alekseyev/alek5.html">Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions</a>. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2
%F a(n) = A014197(n!) = Cardinality[{x; A000010(x) = A000142(n)}].
%e 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 }.
%e From _M. F. Hasler_, Oct 04 2009: (Start)
%e The table A165773 looks as follows:
%e 1,2, (a(1)=2 numbers for which phi(n) = 1! = 1)
%e 3,4,6, (a(2)=3 numbers for which phi(n) = 2! = 2)
%e 7,9,14,18, (a(3)=4 numbers for which phi(n) = 3! = 6)
%e 35,39,45,52,56,70,72,78,84,90, (a(4)=10 numbers for which phi(n) = 4! = 24)
%e ... (End)
%o (Perl) use ntheory ":all"; print "$_ ",scalar(inverse_totient(factorial($_))),"\n" for 1..20; # _Dana Jacobsen_, Mar 04 2019
%Y Cf. A000142, A000010, A014197, A000203, A054873, A067847, A055486, A165774.
%K nonn
%O 1,1
%A _Labos Elemer_, Jun 29 2000
%E More terms from _Jud McCranie_, Jan 02 2001
%E More terms from _David Wasserman_, Apr 30 2002 (with the assistance of _Vladeta Jovovic_ and _Sascha Kurz_)
%E a(21)-a(28) from _Max Alekseyev_, Jan 26 2012, Jul 09 2014
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