A001088 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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

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.

Eugène C. Catalan, Théorèmes de MM. Smith et Mansion, Nouvelle correspondance mathématique (1878) Vol. 4, 103-112.

Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, 76 (2003), 392-394.

Paul 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.

Jorma K. Merikoski, Pentti Haukkanen, Antonio Sasaki, and Timo Tossavainen, On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices, J. Int. Seq. (2025) Vol. 28, Art. 25.7.1.

Henry J. Stephen Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. (1875-1876) Vol. 7, 208-212.

Eric Weisstein's World of Mathematics, Le Paige's Theorem

Index to divisibility sequences

FORMULA

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

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

EXAMPLE

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

The totient function phi(n) is 1, 1, 2, 2, 4, 2, 6 for n = 1 to 7. So a(7) = 1*1*2*2*4*2*6 = 192. - Michael B. Porter, Nov 07 2025

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 *)

PROG

(Haskell)

a001088 n = a001088_list !! (n-1)