A060238 -id:A060238 - 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

A060841 Numerator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).

Original entry on oeis.org

1, 4, 18, 144, 900, 16200, 132300, 2116800, 28576800, 714420000, 8644482000, 311201352000, 4382752374000, 143169910884000, 4026653743612500, 128852919795600000, 2327405863808025000, 125679916645633350000

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), May 02 2001

The value of 1/det(M) is not always an integer! For example, 1/det(35) = 5029296746186844716050163189085401314000634765625/2. - Harry J. Smith, Jul 13 2009
Conjecture: 1/det(M) is an integer only for n: 1 - 34, 36 and 38. All denominators are powers of two (A000079). But not all powers of two are present. See A260502. - Robert G. Wilson v, Aug 02 2015
Values of n at which a(n) = a(n+1): 63, 127, 255, ..., . - Robert G. Wilson v, Aug 03 2015

			a(2) = 4 because the matrix M is [1,1/2; 1/2,1/2] and det(M) = 1/4.

Robert G. Wilson v, Table of n, a(n) for n = 1..400

Cf. A000010, A001088, A060238, A260502, A260897.

d[n_] := Denominator[ Det[ Table[ GCD[1/i, 1/j], {i, n}, {j, n}]]; Array[d, 18]] (* Robert G. Wilson v, Aug 02 2015 *)

vector(20, n, numerator(1/matdet(matrix(n, n, i, j, 1/lcm(i,j))))) \\ Michel Marcus, Aug 03 2015

a(n) = (n!)^2 / (phi(1)*phi(2)*...*phi(n)) = (n!)^2 / A001088(n).

More terms from Reiner Martin, May 17 2001

A060239 a(n) = determinant(P*Q)/n! where P, Q are n X n matrices with P[i,j]=lcm(i,j), Q[i,j]=gcd(i,j).

Original entry on oeis.org

1, -1, 4, -8, 128, 512, -18432, 73728, -884736, -14155776, 1415577600, 11324620800, -1630745395200, -58706834227200, -3757237390540800, 30057899124326400, -7694822175827558400, -92337866109930700800

Offset: 1

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

sign

Enrique Pérez Herrero, Table of n, a(n) for n = 1..100

Cf. A001088, A060238.

def A060239(n):
    P = Matrix(lambda i,j: lcm(i+1,j+1), nrows=n)
    Q = Matrix(lambda i,j: gcd(i+1,j+1), nrows=n)
    return (P*Q).det()/factorial(n) # D. S. McNeil, Jan 16 2011

a(n) = A001088(n)*A060238(n)/n!.

A085905 Permanent of the symmetric n X n matrix M defined by M(i,j) = lcm(i,j) for 1 <= i,j <= n.

Original entry on oeis.org

1, 6, 144, 5952, 772560, 73664640, 29745273600, 8715934402560, 5068085799813120, 2756328707949465600, 4581860819083475558400, 2696083278990328597708800, 7844679216026128507826995200

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003

nonn

			a(2)=6 since the 2 by 2 matrix A with rows [1,2],[2,2] has permanent 1*2+2*2=6.

Vaclav Kotesovec, Table of n, a(n) for n = 1..35

Cf. A060238, A085244, A034444.

with(linalg): a:=(i,j)->lcm(i,j): seq(permanent(matrix(n,n,a)),n=1..14); # Emeric Deutsch, Feb 08 2005

a[n_] := Permanent[Table[LCM[i, j], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 14}] (* Jean-François Alcover, Jan 07 2016 *)

a(n)=matpermanent(matrix(n,n,r,c,lcm(r,c)));
vector(23,n,a(n)) \\ Joerg Arndt, Aug 15 2019

More terms from Emeric Deutsch, Feb 08 2005

A360067 a(n) = det(M) where M is an n X n matrix with M[i,j] = i^j*(i-j).

Original entry on oeis.org

1, 0, 2, 12, 2304, 898560, 4827340800, 143219736576000, 49230909076930560000, 149334225705682285363200000, 5482643392499167214520238080000000, 2322479608280149573505226859610112000000000, 13283541711093841017468807905468592685056000000000000

Offset: 0

José María Grau Ribas, Jan 24 2023

nonn

Cf. A060238, A174890, A176005, A176001, A000178, A089064, A152653.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i,j) -> i^j*(i-j))):
seq(a(n), n=0..12);  # Alois P. Heinz, Jan 25 2023

a[n_] := Det@Table[i^j (i - j), {i, n}, {j, n}]; Table[a[n], {n, 1, 15}]

a(n) = matdet(matrix(n, n, i, j, i^j*(i-j))); \\ Michel Marcus, Jan 24 2023

from sympy import Matrix
def A360067(n): return Matrix(n,n,[i**j*(i-j) for i in range(1,n+1) for j in range(1,n+1)]).det() # Chai Wah Wu, Jan 27 2023

For n>=1, a(n) = A000178(n-1) * A089064(n). - Vaclav Kotesovec, Apr 19 2024

A140412 Determinants of the n X n matrices whose (i,j)-elements are lcm(i^2, j^2).

Original entry on oeis.org

1, -12, 864, -41472, 24883200, 21499084800, -50565847449600, 9708642710323200, -6291200476289433600, -45296643429283921920000, 657707262593202546278400000, 2273036299522107999938150400000, -64536046616031690334243966156800000

Offset: 1

John W. Layman, Jun 17 2008

sign

The determinants of the n X n matrices whose (i,j)-elements are lcm(i,j) are given in A060238.

Cf. A060238.

a(n) = matdet(matrix(n, n, i, j, lcm(i^2, j^2))); \\ Michel Marcus, Jul 10 2014

It appears that a(n) = Product_{k=1..n} MT2(k) * rad(k)^2 * mu(rad(k)), where MT2(k) is the k-th term of the Moebius transform of the sequence of squares, rad(k) is the squarefree kernel of k and mu denotes the Moebius function.

cp's OEIS Frontend

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

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A060841 Numerator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).

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A060239 a(n) = determinant(P*Q)/n! where P, Q are n X n matrices with P[i,j]=lcm(i,j), Q[i,j]=gcd(i,j).

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A085905 Permanent of the symmetric n X n matrix M defined by M(i,j) = lcm(i,j) for 1 <= i,j <= n.

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A360067 a(n) = det(M) where M is an n X n matrix with M[i,j] = i^j*(i-j).

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A140412 Determinants of the n X n matrices whose (i,j)-elements are lcm(i^2, j^2).

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