login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010). 27
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

1,3

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

E. C. Catalan, Théorème de MM. Smith et Mansion, Nouvelle correspondance mathématique, 4 (1878) 103-112. [Philippe Deléham, Dec 22 2003]

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 = 1..496 (first 100 terms by T. D. Noe)

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

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

FORMULA

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

EXAMPLE

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

MAPLE

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

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)

a001088_list = scanl1 (*) a000010_list

-- Reinhard Zumkeller, Mar 04 2012

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

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

CROSSREFS

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

Cf. A003989.

Sequence in context: A269758 A094384 A053038 * A101926 A087965 A074411

Adjacent sequences:  A001085 A001086 A001087 * A001089 A001090 A001091

KEYWORD

nonn,nice,easy

AUTHOR

Simon Plouffe

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)