|
| |
|
|
A165774
|
|
Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010).
|
|
3
| |
|
|
2, 6, 18, 90, 462, 3150, 22050, 210210, 1891890, 19969950, 219669450, 2847714870, 37020293310, 520843112790, 7959363061650, 135309172048050, 2300255924816850, 41996101027370490, 797925919520039310, 16504589035937252250, 347097774991217099850
(list; graph; refs; listen; history; internal format)
|
|
|
|
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)
|
|
|
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: A007869 A144557 A118455 * A053505 A000138 A028857
Adjacent sequences: A165771 A165772 A165773 * A165775 A165776 A165777
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 04 2009
|
|
|
EXTENSIONS
| Edited and terms a(12)-a(21) added by Max Alekseyev (maxale(AT)gmail.com), Jan 26 2012
|
| |
|
|
