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A055506
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Number of solutions to the equation phi(x) = n!.
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6
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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
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1,1
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COMMENTS
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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
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LINKS
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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
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FORMULA
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a(n) = A014197(n!) = Cardinality[{x; A000010(x) = A000142(n)}].
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EXAMPLE
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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)
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PROG
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(Perl) use ntheory ":all"; print "$_ ", scalar(inverse_totient(factorial($_))), "\n" for 1..20; # Dana Jacobsen, Mar 04 2019
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Labos Elemer, Jun 29 2000
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EXTENSIONS
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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
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STATUS
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approved
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