A001088 -id:A001088 - cp's OEIS frontend

A322175 Determinant of the symmetric n X n matrix M defined by M(i,j) = mu(gcd(i,j)) for 1 <= i,j <= n where mu is the Moebius function.

1, 1, -2, 4, 4, -8, -32, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Bernard Schott, Dec 02 2018

sign

a(n) <> 0 for 0 <= n <= 7, but a(n) = 0 for n >= 8.

			For n = 2,
               [ mu(1)  mu(1) ]     [ 1  1 ]
the matrix is  [              ]  =  [      ]
               [ mu(1)  mu(2) ]     [ 1 -1 ]
so a(2) = -2.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 694 pp. 90, 297, Ellipses Paris 2004.

Cf. A008683, A001088 (determinant of n X n matrix M with M(i,j) = gcd(i,j))

m[i_,j_] := MoebiusMu[GCD[i,j]]; a[n_] := Det[Table[m[i,j], {i,1,n}, {j,1,n}]]; Array[a, 30] (* Amiram Eldar, Dec 02 2018 *)

a(n) = matdet(matrix(n, n, i, j, moebius(gcd(i,j)))); \\ Michel Marcus, Dec 03 2018

A330967 a(n) is the determinant of the matrix with elements gcd(i,j) for 2 <= i,j <= n.

Original entry on oeis.org

2, 5, 10, 44, 104, 656, 2624, 15744, 67584, 694272, 2777088, 34062336, 213221376, 1758855168, 14070841344, 228530847744, 1371185086464, 25007480635392, 200059845083136, 2447683608379392, 25040421692375040, 556525133318062080, 4452201066544496640, 89044021330889932800

Offset: 2

Matt Frank, Jan 04 2020

nonn

These determinants are always nonzero, as shown by Beslin and Ligh.

Scott Beslin and Steven Ligh, Greatest Common Divisor Matrices, Linear Algebra and Its Applications 119 (1989), 69-76.

A001088 gives the determinants for gcd(i,j), 1 <= i,j <= n.

A067549 gives the determinants for gcd(i-th prime, j-th prime), 1 <= i,j <= n.

Table[Det[Table[GCD[i, j], {i, 2, n}, {j, 2, n}]], {n, 2, 25}]

a(n)={matdet(matrix(n-1, n-1, i, j, gcd(i+1,j+1)))} \\ Andrew Howroyd, Jan 07 2020

A344786 Decimal expansion of (1/e) * Product_{p prime} (1 - 1/p)^(1/p).

Original entry on oeis.org

2, 0, 5, 9, 6, 3, 0, 5, 0, 2, 8, 8, 1, 8, 6, 3, 5, 3, 8, 7, 9, 6, 7, 5, 4, 2, 8, 2, 3, 2, 4, 9, 7, 4, 6, 6, 4, 8, 5, 8, 7, 8, 0, 5, 9, 3, 4, 2, 0, 5, 8, 5, 1, 5, 0, 1, 6, 4, 2, 7, 8, 8, 1, 5, 1, 3, 6, 5, 7, 4, 9, 3, 0, 9, 9, 4, 3, 5, 4, 7, 6, 6, 3, 8, 1, 2, 4

Offset: 0

Amiram Eldar, May 28 2021

Deshouillers and Iwaniec (2008) proved that the sequence of geometric mean values of the Euler totient function, A001088(n)^(1/n) = (Product_{k=1..n} phi(k))^(1/n), is uniformly distributed modulo 1 if and only if this constant is irrational. They noted that Richard Bumby showed that if it is rational, then its denominator has at least 20 decimal digits.

			0.20596305028818635387967542823249746648587805934205...

Jean-Marc Deshouillers and Henryk Iwaniec, On the distribution modulo one of the mean values of some arithmetical functions, Uniform Distribution Theory, Vol. 3, No. 1 (2008), pp. 111-124.
Mehdi Hassani, Uniform distribution modulo one of some sequences concerning the Euler function, Rev. Un. Mat. Argentina, Vol. 54, No. 1 (2013), pp. 55-68.

Cf. A000010, A001088, A068985.

$MaxExtraPrecision = 1000; m = 100; RealDigits[N[Exp[-1 - Sum[PrimeZetaP[k]/(k - 1), {k, 2, 1000}]], m + 1], 10, m][[1]]

Equals exp(-1 - Sum_{k>=2} P(k)/(k-1)), where P(s) is the prime zeta function.

cp's OEIS Frontend

A322175 Determinant of the symmetric n X n matrix M defined by M(i,j) = mu(gcd(i,j)) for 1 <= i,j <= n where mu is the Moebius function.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

A330967 a(n) is the determinant of the matrix with elements gcd(i,j) for 2 <= i,j <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A344786 Decimal expansion of (1/e) * Product_{p prime} (1 - 1/p)^(1/p).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula