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A165774 Largest solution to phi(x) = n!, where phi() is Euler totient function (A000010). 4
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 (list; graph; refs; listen; history; text; 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)

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

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Last modified February 1 06:22 EST 2023. Contains 359981 sequences. (Running on oeis4.)