A260897 -id:A260897 - cp's OEIS frontend

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

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

A260502 Log_2 of the numerator of det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 4, 5, 3, 6, 5, 6, 2, 3, 3, 8, 7, 9, 8, 11, 8, 10, 10, 11, 11, 13, 12, 14, 7, 11, 11, 12, 13, 15, 16, 17, 14, 17, 17, 20, 18, 20, 21, 22, 19, 20, 21, 22, 21, 27, 26, 29, 26, 29

Offset: 1

Robert G. Wilson v, Aug 02 2015

nonn

Powers of two not present in A260897: 23, 24, 25, 28, 38, 46, 47, 49, 55, 63, 64, 69, ..., .

			a(4) = 0 because for n=4 det(M) = 1/144.
a(35) = 1 because for n=35 det(M) equals 2/5029296746186844716050163189085401314000634765625.

Robert G. Wilson v, Table of n, a(n) for n = 1..500

Cf. A060841, A007814, A260897.

f[n_] := Log2@ Numerator@ Det@ Table[ 1/LCM[i, j], {i, n}, {j, n}]; Array[f, 85]

vector(80, n, valuation(denominator(1/matdet(matrix(n, n, i, j, 1/lcm(i, j)))), 2)) \\ Michel Marcus, Aug 04 2015

a(n) = A007814(A260897(n)).

A260909 Denominator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/gcd(i,j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 4, 32, 2, 32, 32, 64, 64, 256, 256, 512, 128, 512, 1024, 2048, 1024, 4096, 16384, 32768, 1024, 4096, 32768, 262144, 131072, 524288, 524288, 4194304, 2097152, 16777216, 33554432, 67108864, 33554432, 268435456, 268435456

Offset: 1

Robert G. Wilson v, Aug 04 2015

nonn

All terms are powers of two (A000079).

Cf. A000079, A260897, A260908.

seq(denom(1/LinearAlgebra:-Determinant(Matrix(n,n,1/igcd))),n=1..100); # Robert Israel, Aug 16 2015

f[n_] := 1/Det[ Table[ 1/GCD[i, j], {i, n}, {j, n}]]; Denominator@ Array[f, 46]

vector(50, n, denominator(1/matdet(matrix(n, n, i, j, 1/gcd(i, j))))) \\ Michel Marcus, Aug 07 2015

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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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A260502 Log_2 of the numerator of det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).

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A260909 Denominator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/gcd(i,j).

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