A055489 -id:A055489 - 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

A055486 Number of solutions to sigma(x) = n!.

Original entry on oeis.org

1, 0, 1, 3, 4, 15, 33, 111, 382, 1195, 3366, 14077, 53265, 229603, 910254, 4524029, 18879944, 91336498, 561832582, 2801857644, 14652294729, 78894985156, 408373652461, 2378940665083, 11939275822636, 71931330299023, 392274481206066, 2626331088771946

Offset: 1

Labos Elemer, Jun 28 2000

nonn

			For n = 9, solutions to sigma(x) = n! = 362880 form a set {97440, ..., 361657} of size 382, so a(9) = 382.

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. A000142, A000203, A054973, A014197, A055488, A055489, A055506.

with(numtheory): for f from 1 to 9 do fac := f!: k := 0:for n from 1 to fac do if sigma(n)=fac then k := k+1: fi: od: print( k); od:

a(n) = A054973(n!) = Cardinality[{x; A000203(x) = A000142(n)}].

More terms from Jud McCranie, Oct 09 2000
a(13)-a(14) from Donovan Johnson, Nov 22 2008
a(15) from Ray Chandler, Jan 13 2009
a(16)-a(28) from Max Alekseyev, Jan 23 2012

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

A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

Original entry on oeis.org

2, 6, 18, 90, 462, 3150, 22050, 210210, 1891890, 19969950, 219669450, 2847714870, 37020293310, 520843112790, 7959363061650, 135309172048050, 2300255924816850, 41996101027370490, 797925919520039310, 16504589035937252250, 347097774991217099850, 7751850308137181896650, 179602728970220622816750, 4493489228616853106091450, 112337230715421327652286250, 2958213742172761628176871250, 79871771038664563960775523750, 2279417465795734863803670716250

Offset: 1

M. F. Hasler, Oct 04 2009

nonn

All solutions to phi(x) = n! belong to the interval [n!,(n+1)!] and are listed in the n-th row of A165773 (when written as table with row lengths A055506). Thus this sequence gives the last element in these rows, i.e., a(n) = A165773(Sum_{k=1..n} A055506(k)).
All terms in this sequence are even, since if x is an odd solution to phi(x) = n!, then 2x is a larger solution because phi(2x) = phi(2)*phi(x) = phi(x).
Most terms (and any term divisible by 4) are divisible by 3, since if x = 2^k*y is a solution with k>1 and gcd(y,2*3) = 1, then x*3/2 = 2^(k-1)*3*y is a larger solution because phi(2^(k-1)*3) = 2^(k-2)*(3-1) = 2^(k-1) = phi(2^k).
For the same reason, most terms are divisible by 5, since if x=2^k*y is a solution with k>2 and gcd(y,2*5) = 1, then x*5/4 is a larger solution.
Also, any term of the form x = 2^k*3^m*y with k,m>1 must be divisible by 7 (else x*7/6 would be a larger solution), and so on.
Experimentally, a(n) = c(n)*(n+1)! with a coefficient c(n) ~ 2^(-n/10) (e.g., c(1) = c(2) = 1, c(10) ~ 0.5).

			a(1) = 2 is the largest among the A055506(1) = 2 solutions {1,2} to phi(n) = 1! = 1.
a(4) = 90 is the largest among the A055506(4) = 10 solutions {35, 39, 45, 52, 56, 70, 72, 78, 84, 90} to phi(n) = 4! = 24.
See A165773 for more examples.

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, A055487, A055506, A055489, A014197, A165773.

a(n) = invphiMax(n!); \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Edited and terms a(12)-a(28) added by Max Alekseyev, Jan 26 2012, Jul 09 2014

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

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A055486 Number of solutions to sigma(x) = n!.

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

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A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).

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