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

%I A001088 #86 Feb 16 2025 08:32:22
%S A001088 1,1,1,2,4,16,32,192,768,4608,18432,184320,737280,8847360,53084160,
%T A001088 424673280,3397386240,54358179840,326149079040,5870683422720,
%U A001088 46965467381760,563585608581120,5635856085811200,123988833887846400,991910671102771200,19838213422055424000
%N A001088 Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).
%C A001088 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 A001088 The matrix M(i,j) = gcd(i,j) is sequence A003989. - _Michael Somos_, Jun 25 2012
%D A001088 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 598.
%D A001088 M. Petkovsek et al., A=B, Peters, 1996, p. 21.
%H A001088 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 A001088 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 A001088 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 A001088 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 A001088 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 A001088 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 A001088 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 A001088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LePaigesTheorem.html">Le Paige's Theorem</a>
%H A001088 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A001088 a(n) = phi(1) * phi(2) * ... * phi(n).
%F A001088 Limit_{n->infinity} a(n)^(1/n) / n  = exp(-1) * A124175 = 0.205963050288186353879675428232497466485878059342058515016427881513657493... (see Mathoverflow link). - _Vaclav Kotesovec_, Jun 09 2021
%e A001088 a(2) = 1 because the matrix M is: [1,1; 1,2] and det(A) = 1.
%p A001088 with(numtheory,phi); A001088 := proc(n) local i; mul(phi(i),i=1..n); end;
%p A001088 seq(A001088(n), n=0..30);
%t A001088 A001088[n_]:=Times@@EulerPhi/@Range[n]; Table[A001088[n], {n, 30}] (* _Enrique Pérez Herrero_, Sep 19 2010 *)
%t A001088 Rest[FoldList[Times,1,EulerPhi[Range[30]]]] (* _Harvey P. Dale_, Dec 09 2011 *)
%o A001088 (Haskell)
%o A001088 a001088 n = a001088_list !! (n-1)
%o A001088 a001088_list = scanl1 (*) a000010_list
%o A001088 -- _Reinhard Zumkeller_, Mar 04 2012
%o A001088 (PARI) a(n)=prod(k=1,n,eulerphi(k)) \\ _Charles R Greathouse IV_, Mar 04 2012
%o A001088 (GAP) List([1..30],n->Product([1..n],i->Phi(i))); # _Muniru A Asiru_, Jul 31 2018
%Y A001088 Cf. A000010, A060238, A060239, A059381, A059382, A059383, A059384, A002088.
%Y A001088 Cf. A003989.
%K A001088 nonn,nice,easy
%O A001088 0,4
%A A001088 _Simon Plouffe_
%E A001088 a(0)=1 prepended by _Alois P. Heinz_, Jul 19 2023