cp's OEIS Frontend

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

%I A055506 #55 Nov 15 2024 06:57:29
%S A055506 2,3,4,10,17,49,93,359,1138,3802,12124,52844,182752,696647,2852886,
%T A055506 16423633,75301815,367900714,1531612895,8389371542,40423852287,
%U A055506 213232272280,1295095864798,7991762413764,42259876674716,252869570952706,1378634826630301,8749244047999717
%N A055506 Number of solutions to the equation phi(x) = n!.
%C A055506 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 A055506 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 A055506 Andrew Lelechenko, <a href="/A055506/b055506.txt">Table of n, a(n) for n = 1..36</a>
%H A055506 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.
%H A055506 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%F A055506 a(n) = A014197(n!) = Cardinality({x; A000010(x) = A000142(n)}).
%e A055506 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 A055506 From _M. F. Hasler_, Oct 04 2009: (Start)
%e A055506 The table A165773 looks as follows:
%e A055506   1,2, (a(1)=2 numbers for which phi(n) = 1! = 1)
%e A055506   3,4,6, (a(2)=3 numbers for which phi(n) = 2! = 2)
%e A055506   7,9,14,18, (a(3)=4 numbers for which phi(n) = 3! = 6)
%e A055506   35,39,45,52,56,70,72,78,84,90, (a(4)=10 numbers for which phi(n) = 4! = 24)
%e A055506   ... (End)
%o A055506 (Perl) use ntheory ":all"; print "$_ ",scalar(inverse_totient(factorial($_))),"\n" for 1..20; # _Dana Jacobsen_, Mar 04 2019
%o A055506 (PARI) a(n) = invphiNum(n!); \\ _Amiram Eldar_, Nov 15 2024 using _Max Alekseyev_'s invphi.gp
%Y A055506 Cf. A000010, A000142, A000203, A014197, A054873, A055486, A067847, A165773, A165774.
%K A055506 nonn
%O A055506 1,1
%A A055506 _Labos Elemer_, Jun 29 2000
%E A055506 More terms from _Jud McCranie_, Jan 02 2001
%E A055506 More terms from _David Wasserman_, Apr 30 2002 (with the assistance of _Vladeta Jovovic_ and _Sascha Kurz_)
%E A055506 a(21)-a(28) from _Max Alekseyev_, Jan 26 2012, Jul 09 2014