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

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

1,1

Comments

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(A055506(k),k=1..n)).

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)

Links

Table of n, a(n) for n=1..28.

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

Example

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.

Crossrefs

Cf. A055487, A055506, A055489, A014197

Sequence in context: A327729 A273001 A118455 * A053505 A000138 A028857

Adjacent sequences: A165771 A165772 A165773 * A165775 A165776 A165777

Keyword

nonn

Author

M. F. Hasler, Oct 04 2009

Extensions

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

Status

approved