A048855 Number of integers up to n! relatively prime to n!.

1, 1, 1, 2, 8, 32, 192, 1152, 9216, 82944, 829440, 8294400, 99532800, 1194393600, 16721510400, 250822656000, 4013162496000, 64210599936000, 1155790798848000, 20804234379264000, 416084687585280000, 8737778439290880000, 192231125664399360000

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, D. 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

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

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

Cf. A000010, A000142, A014197.

Sequence in context: A081358 A294506 A206303 * A262480 A062797 A369645

Adjacent sequences: A048852 A048853 A048854 * A048856 A048857 A048858

Keyword

easy,nonn

Author

Paul Max Payton

Extensions

Name changed by Daniel Forgues, Aug 01 2011

Status

approved