OFFSET
0,4
COMMENTS
Rephrasing the Quet formula: Begin with 1. Then, if n + 1 is prime subtract 1 and multiply. If n+1 is not prime, multiply. Continue writing each product. Thus the sequence would begin 1, 2, 8, . . . . The first product is 1*(2 - 1), second is 1*(3 - 1), and third is 2*4. - Enoch Haga, May 06 2009
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, page 134.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..450
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.
FORMULA
a(n) = phi(n!) = A000010(n!).
If n is composite, then a(n) = a(n-1)*n. If n is prime, then a(n) = a(n-1)*(n-1). - Leroy Quet, May 24 2007
Under the Riemann Hypothesis, a(n) = n! / (e^gamma * log n) * (1 + O(log n/sqrt(n))). - Charles R Greathouse IV, May 12 2011
Sum_{k=1..n} a(k) = exp(-gamma) * (n!/log(n)) * (1 + O(1/log(n)^3)), where gamma is Euler's constant (A001620) (De Koninck and Verreault, 2024, p. 56, eq. (4.12)). - Amiram Eldar, Dec 10 2024
MAPLE
with(numtheory):a:=n->phi(n!): seq(a(n), n=0..20); # Zerinvary Lajos, Oct 07 2007
MATHEMATICA
Table[ EulerPhi[ n! ], {n, 0, 21}] (* Robert G. Wilson v, Nov 21 2003 *)
PROG
(Sage) [euler_phi(factorial(n)) for n in range(0, 21)] # Zerinvary Lajos, Jun 06 2009
(PARI) a(n)=eulerphi(n!) \\ Charles R Greathouse IV, May 12 2011
(Python)
from math import factorial, prod
from sympy import primerange
from fractions import Fraction
def A048855(n): return (factorial(n)*prod(Fraction(p-1, p) for p in primerange(n+1))).numerator # Chai Wah Wu, Jul 06 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
Name changed by Daniel Forgues, Aug 01 2011
STATUS
approved