A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

1, 1, 1, 2, 4, 16, 32, 192, 768, 4608, 18432, 184320, 737280, 8847360, 53084160, 424673280, 3397386240, 54358179840, 326149079040, 5870683422720, 46965467381760, 563585608581120, 5635856085811200, 123988833887846400, 991910671102771200, 19838213422055424000

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• nice
• easy

Comments

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n [Smith and Mansion]. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001

The matrix M(i,j) = gcd(i,j) is sequence A003989. - Michael Somos, Jun 25 2012

Examples

			a(2) = 1 because the matrix M is: [1,1; 1,2] and det(A) = 1.
		

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.
• M. Petkovsek et al., A=B, Peters, 1996, p. 21.

Links

• Antoine Mathys, Table of n, a(n) for n = 0..496 (first 100 terms by T. D. Noe)
• Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
• E. C. Catalan, Théorème de MM. Smith et Mansion, Nouvelle correspondance mathématique, 4 (1878) 103-112. [_Philippe Deléham_, Dec 22 2003]
• Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, 76 (2003), 392-394.
• P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
• Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
• H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
• Eric Weisstein's World of Mathematics, Le Paige's Theorem
• Index to divisibility sequences

Crossrefs

Cf. A000010, A060238, A060239, A059381, A059382, A059383, A059384, A002088.

Cf. A003989.

Programs

GAP

List([1..30],n->Product([1..n],i->Phi(i))); # Muniru A Asiru, Jul 31 2018

Haskell

a001088 n = a001088_list !! (n-1)
a001088_list = scanl1 (*) a000010_list
-- Reinhard Zumkeller, Mar 04 2012

Maple

with(numtheory,phi); A001088 := proc(n) local i; mul(phi(i),i=1..n); end;
seq(A001088(n), n=0..30);

Mathematica

A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* Enrique Pérez Herrero, Sep 19 2010 *)
Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* Harvey P. Dale, Dec 09 2011 *)

PARI

a(n)=prod(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Mar 04 2012

Formula

a(n) = phi(1) * phi(2) * ... * phi(n).

Limit_{n->infinity} a(n)^(1/n) / n = exp(-1) * A124175 = 0.205963050288186353879675428232497466485878059342058515016427881513657493... (see Mathoverflow link). - Vaclav Kotesovec, Jun 09 2021

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 19 2023