A001088 - OEIS

%I #85 Aug 05 2023 06:26:53

%S 1,1,1,2,4,16,32,192,768,4608,18432,184320,737280,8847360,53084160,

%T 424673280,3397386240,54358179840,326149079040,5870683422720,

%U 46965467381760,563585608581120,5635856085811200,123988833887846400,991910671102771200,19838213422055424000

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

%C 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

%C The matrix M(i,j) = gcd(i,j) is sequence A003989. - _Michael Somos_, Jun 25 2012

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.

%D M. Petkovsek et al., A=B, Peters, 1996, p. 21.

%H Antoine Mathys, <a href="/A001088/b001088.txt">Table of n, a(n) for n = 0..496</a> (first 100 terms by T. D. Noe)

%H Antal Bege, <a href="http://www.emis.de/journals/AUSM/C1-1/MATH1-4.PDF">Hadamard product of GCD matrices</a>, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

%H E. C. Catalan, <a href="https://gdz.sub.uni-goettingen.de/id/PPN598948236_0004?tify=%7B%22pages%22%3A%5B121%5D%2C%22view%22%3A%22info%22%7D">Théorème de MM. Smith et Mansion</a>, Nouvelle correspondance mathématique, 4 (1878) 103-112. [_Philippe Deléham_, Dec 22 2003]

%H Warren P. Johnson, <a href="http://www.jstor.org/stable/3654887">An LDU Factorization in Elementary Number Theory</a>, Mathematics Magazine, 76 (2003), 392-394.

%H P. Mansion, <a href="https://archive.org/stream/messengermathem01glaigoog#page/n94/mode/2up">On an Arithmetical Theorem of Professor Smith's</a>, Messenger of Mathematics, (1878), pp. 81-82.

%H Mathoverflow, <a href="http://mathoverflow.net/questions/230318/asymptotics-of-product-of-eulers-totient-function-a001088">Asymptotics of product of Euler's totient function</a>, 2016.

%H H. J. S. Smith, <a href="http://plms.oxfordjournals.org/content/s1-7/1/208.extract">On the value of a certain arithmetical determinant</a>, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LePaigesTheorem.html">Le Paige's Theorem</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

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

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

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

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

%p seq(A001088(n), n=0..30);

%t A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* _Enrique Pérez Herrero_, Sep 19 2010 *)

%t Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* _Harvey P. Dale_, Dec 09 2011 *)

%o (Haskell)

%o a001088 n = a001088_list !! (n-1)

%o a001088_list = scanl1 (*) a000010_list

%o -- _Reinhard Zumkeller_, Mar 04 2012

%o (PARI) a(n)=prod(k=1,n,eulerphi(k)) \\ _Charles R Greathouse IV_, Mar 04 2012

%o (GAP) List([1..30],n->Product([1..n],i->Phi(i))); # _Muniru A Asiru_, Jul 31 2018

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

%Y Cf. A003989.

%K nonn,nice,easy

%O 0,4

%A _Simon Plouffe_

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 19 2023