A203415 Vandermonde determinant of the first n nonprimes (A018252).

1, 3, 30, 1680, 201600, 87091200, 1103619686400, 275463473725440000, 240529195987579699200000, 1163776461866305616609280000000, 344605941225348705438623229542400000000, 3717059729911125118574880410324812431360000000000

Offset: 1

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A203416, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203417.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A000040, A018252, A203416, A203417.

Programs

Magma

A018252:=[n : n in [1..250] | not IsPrime(n) ];
A203415:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;
[A203415(n): n in [1..30]]; // G. C. Greubel, Feb 29 2024

Mathematica

z=20;
nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
f[j_]:= nonprime[[j]];
v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,1,n}];
Table[v[n], {n,1,z}]             (* this sequence *)
Table[v[n+1]/v[n], {n,1,z}]      (* A203416 *)
Table[v[n]/d[n], {n,1,z}]        (* A203417 *)

SageMath

A018252=[n for n in (1..250) if not is_prime(n)]
def A203415(n): return product(product(A018252[k+1]-A018252[j] for j in range(k+1)) for k in range(n-1))
[A203415(n) for n in range(1,31)] # G. C. Greubel, Feb 29 2024