A060239 -id:A060239 - cp's OEIS frontend

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

Simon Plouffe

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

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

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.

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

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

Cf. A003989.

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

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

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

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 *)

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

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

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

A060238 a(n) = det(M) where M is an n X n matrix with M[i,j] = lcm(i,j).

Original entry on oeis.org

1, 1, -2, 12, -48, 960, 11520, -483840, 3870720, -69672960, -2786918400, 306561024000, 7357464576000, -1147764473856000, -96412215803904000, -11569465896468480000, 185111454343495680000, -50350315581430824960000, -1812611360931509698560000

Offset: 0

MCKAY john (mckay(AT)cs.concordia.ca), Mar 21 2001

sign

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 695, pp. 90, 297-298, Ellipses, Paris, 2004.
J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, p. 265, eq. 10.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..200
Index to sequences related to determinants.

Cf. A000142, A001088, A013939, A023900, A048803, A060239, A085542.

A060238:=n->n!*mul((1-ithprime(i))^floor(n/ithprime(i)), i=1..numtheory[pi](n)): seq(A060238(n), n=0..20); # Wesley Ivan Hurt, Aug 15 2016

A060238[n_]:=n!*Product[(1 - Prime[i])^Floor[n/Prime[i]], {i, PrimePi[n]}]; Array[A060238, 20] (* Enrique Pérez Herrero, Jun 08 2010 *)

a(n)=n!*prod(p=1,sqrtint(n),if(isprime(p),(1-p)^floor(n/p),1)) \\ Benoit Cloitre, Jan 31 2008

For n >= 2, a(n) = n! * Product_{j=2..n} Product_{p|j} (1-p) (where the second product is over all primes p that divide j) (cf. A023900). - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001
a(n) = n! * Product_{p<=n} (1-p)^floor(n/p) where the product runs through the primes. - Benoit Cloitre, Jan 31 2008
a(n) = A000142(n) * A085542(n). - Enrique Pérez Herrero, Jun 08 2010
a(n) = A001088(n) * A048803(n) * (-1)^A013939(n). - Amiram Eldar, Dec 19 2018
a(n) = Product_{k=1..n} (-1)^A001221(k) * A000010(k) * A007947(k) [De Koninck & Mercier]. - Bernard Schott, Dec 11 2020

a(0)=1 prepended by Alois P. Heinz, Jan 25 2023

cp's OEIS Frontend

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

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