A055486 -id:A055486 - cp's OEIS frontend

A055487 Least m such that phi(m) = n!.

1, 3, 7, 35, 143, 779, 5183, 40723, 364087, 3632617, 39916801, 479045521, 6227180929, 87178882081, 1307676655073, 20922799053799, 355687465815361, 6402373865831809, 121645101106397521, 2432902011297772771, 51090942186005065121, 1124000727844660550281, 25852016739206547966721, 620448401734814833377121, 15511210043338862873694721, 403291461126645799820077057, 10888869450418352160768000001, 304888344611714964835479763201

Offset: 1

Labos Elemer, Jun 28 2000

nonn

Erdős believed (see Guy reference) that phi(x) = n! is solvable.
Factorial primes of the form p = A002981(m)! + 1 = k! + 1 give the smallest solutions for some m (like m = 1,2,3,11) as follows: phi(p) = p-1 = A002981(m)!.
According to Tattersall, in 1950 H. Gupta showed that phi(x) = n! is always solvable. - Joseph L. Pe, Oct 01 2002
A123476(n) is a solution to the equation phi(x)=n!. - T. D. Noe, Sep 27 2006
From M. F. Hasler, Oct 04 2009: (Start)
Conjecture: Unless n!+1 is prime (i.e., n in A002981), a(n)=pq where p is the least prime > sqrt(n!) such that (p-1) | n! and q=n!/(p-1)+1 is prime.
Probably "least prime > sqrt(n!)" can also be replaced by "largest prime <= ceiling(sqrt(n!))". The case "= ceiling(...)" occurs for n=5, sqrt(120) = 10.95..., p=11, q=13.
a(n) is the first element in row n of the table A165773, which lists all solutions to phi(x)=n!. Thus a(n) = A165773((Sum_{kA055506(k)) + 1). The last element of each row (i.e., the largest solution to phi(x)=n!) is given in A165774. (End)

R. K. Guy, (1981): Unsolved problems In Number Theory, Springer - page 53.
Tattersall, J., "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, p. 162.

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.
P. Erdős and J. Lambek, Problem 4221, Amer. Math. Monthly, 55 (1948), 103.

Cf. A055486, A055488, A055489, A055506, A000010, A000142.

Cf. A123476, A165773, A165774.

Array[Block[{k = 1}, While[EulerPhi[k] != #, k++]; k] &[#!] &, 10] (* Michael De Vlieger, Jul 12 2018 *)

a(n) = Min{m : phi(m) = n!} = Min{m : A000010(m) = A000142(n)}.

More terms from Don Reble, Nov 05 2001

a(21)-a(28) from Max Alekseyev, Jul 09 2014

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

Original entry on oeis.org

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

Labos Elemer, Jun 29 2000

nonn

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

			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)

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).

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

a(n) = invphiNum(n!); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

use ntheory ":all"; print "$ ",scalar(inverse_totient(factorial($))),"\n" for 1..20; # Dana Jacobsen, Mar 04 2019

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

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

A055489 Largest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 23, 95, 719, 5039, 39917, 361657, 3624941, 39904153, 479001599, 6226862869, 87178291199, 1307672080867, 20922780738961, 355687390376431, 6402373545694717, 121645099711277873, 2432902005056589697, 51090942157413850441

Offset: 3

Labos Elemer, Jun 28 2000

nonn

For n = 1, a(1) = 1; for n = 2 there is no solution.

For n in A002982, a(n) = n!-1.

			For n = 6, the 15 solutions are as follows: {264, 270, 280, 354, 376, 406, 418, 435, 459, 478, 537, 623, 649, 667, 719}.

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

Ray Chandler, Table of n, a(n) for n=3..52
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.

Cf. A000203, A000142, A002982, A055486, A055488, A057637.

a(n) = {fn = n!; x = fn - 1; while(sigma(x) != fn, x--); x;} \\ Michel Marcus, Dec 17 2013

a(n) = Max{x; Sigma[x] = n!} = Max{x; A000203(x) = A000142(n)}
a(n) = A057637(A000142(n)). - Ray Chandler, Jan 15 2009
a(A002982(n)) = A000142(A002982(n)) - 1. - Ray Chandler, Jan 15 2009

More terms from Jud McCranie, Oct 09 2000
a(15) from Donovan Johnson, Aug 31 2008
a(16)-a(19) from Donovan Johnson, Nov 22 2008
a(20)-a(52) from Ray Chandler, Jan 15 2009

A055488 Smallest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 14, 54, 264, 1560, 10920, 97440, 876960, 10263240, 112895640, 1348827480, 18029171160, 264370186080, 3806158356000, 62703141621120, 1128159304272000, 20422064875212000, 404757215566704000, 8208550091549808000, 177650747421074928000

Offset: 3

Labos Elemer, Jun 28 2000

nonn

For n = 1, a(1) = 1; for n = 2, there is no solution.

			For n = 6 the 15 solutions are as follows: {264,270,280,354,376,406,418,435,459,478,537,623,649,667,719}

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

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

Cf. A000203, A000142, A055486, A055487, A055489

a(n) = Min{x; Sigma[x] = n!} = Min{x; A000203(x) = A000142(n)}

More terms from Jud McCranie, Oct 09 2000
a(13), a(14) from Vim Wenders, Nov 06 2006, Jan 12 2007
a(15), a(16) from Donovan Johnson, Aug 26 2008, Mar 26 2010
a(17)-a(22) from Max Alekseyev, Jan 25 2012

A245015 Numbers n such that sigma(n) is a factorial.

Original entry on oeis.org

1, 5, 14, 15, 23, 54, 56, 87, 95, 264, 270, 280, 354, 376, 406, 418, 435, 459, 478, 537, 623, 649, 667, 719, 1560, 1740, 1824, 1836, 1992, 2148, 2214, 2262, 2296, 2392, 2470, 2492, 2514, 2528, 2596, 2668, 2876, 3058, 3154, 3471, 3567, 3777, 3835, 3895, 4147, 4195, 4199

Offset: 1

Alex Ratushnyak, Sep 16 2014

nonn

Numbers n such that A000203(n) is in A000142.

			sigma(5) is 6 which is 3!, so 5 is n the sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000203, A000142, A055486.

Select[Range[4000], MemberQ[Range[8]!, DivisorSigma[1, #]] &] (* Alonso del Arte, Sep 24 2014 *)

isok(n) = {if (n==1, return (1)); s = sigma(n); f = 1; for (k=2, s, f *= k; if (f == s, return (1)); if (f > s, return (0)););} \\ Michel Marcus, Mar 09 2015

A354074 Factorials that are the sum of the divisors of some m.

Original entry on oeis.org

1, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

Sequence of different values of A000203(A245015(n)).

Conjecture: number 2 is the only factorial that is not in this sequence.

			Number 24 is in the sequence because sigma(14) = sigma(15) = sigma(23) = 24.

Intersection of A000142 and A002191.

Cf. A000203, A338112, A055486, A055488, A245015, A354072, A354073, A338112.

More terms from Jinyuan Wang, May 17 2022

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A055487 Least m such that phi(m) = n!.

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

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A055489 Largest number x such that sum of divisors of x is n!.

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A055488 Smallest number x such that sum of divisors of x is n!.

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A245015 Numbers n such that sigma(n) is a factorial.

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A354074 Factorials that are the sum of the divisors of some m.

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