A080358 Value of Vandermonde determinant for the first n prime numbers: V[prime(1), ..., prime(n)].

1, 1, 1, 6, 240, 414720, 4379443200, 2648687247360000, 11619303595714805760000, 4047756373260469165621248000000, 311107430628520522709128328175943680000000, 152539657943794787580793302587123569672794931200000000

Offset

0,4

Comments

The value of the Vandermonde determinant is unchanged if the numbers are shifted by an arbitrary constant c, i.e., V[prime(1), ..., prime(n)] = V[prime(1)-c, ..., prime(n)-c].

For a guide to related sequences, see A093883. - Clark Kimberling, Jan 03 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 0..36

Formula

a(n) = Product_{i, j, i>j} (prime(i) - prime(j)). a(n) is the product of binomial(n, 2) prime differences of not necessarily consecutive primes.

For n > 1, a(n) = sqrt(Delta_n), where Delta_n is the discriminant of the polynomial (x - 2)*(x - 3)*...*(x - prime(n)). - Thomas Ordowski, Mar 15 2023

Example

a(1)=1 corresponds to 1 X 1 V-matrix, while a(2)=1 is computed from a 2 X 2 matrix.
n = 2: a(2) = prime(2) - prime(1) = 3 - 2 = 1;
n = 3: a(3) = (5-3)*(5-2)*(3-2) = 2*3*1 = 6; ...
n = 6: a(6) = (13-11)*(13-7)*(13-5)*(13-3)*(13-2)*(11-7)*(11-5)*(11-3)*(11-2)*(7-5)*(7-3)*(7-2)*(5-3)*(5-2)*(3-2) = 2*6*8*10*11*4*6*8*9*2*4*5*2*3*1 = 10560*1728*40*6*1 = 4379443200.

Maple

with(LinearAlgebra):
a:= n-> Determinant(Matrix(n, (i, j)-> ithprime(i)^(j-1))):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 22 2017

Mathematica

b[x_] := Prime[x] d[x_] := b[x+1]-b[x] t[m_] := b[m+1]-Table[b[x], {x, 1, m}] pt[x_] := Apply[Times, t[x]] va[x_] := Apply[Times, Table[pt[w], {w, 1, x}]] Table[va[j], {j, 1, 10}]

Crossrefs

Cf. A000040, A001359, A203521, A290179.

Sequence in context: A234633 A137892 A064382 * A233826 A156936 A231019

Adjacent sequences: A080355 A080356 A080357 * A080359 A080360 A080361

Keyword

nonn

Author

Labos Elemer, Feb 19 2003

Status

approved