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Vandermonde determinant of the first n composite numbers (A002808).
6

%I #12 Feb 24 2024 11:03:52

%S 1,2,16,240,11520,13271040,254803968000,15892123484160000,

%T 5126163351050649600000,89288743527804466888704000000,

%U 50689719717698351557731837542400000000,125765178831579421305165126665125232640000000000

%N Vandermonde determinant of the first n composite numbers (A002808).

%C Each term divides its successor, as in A203419, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203420.

%H G. C. Greubel, <a href="/A203418/b203418.txt">Table of n, a(n) for n = 1..42</a>

%t composite = Select[Range[100], CompositeQ]; (* A002808 *)

%t z = 20;

%t f[j_]:= composite[[j]];

%t v[n_]:= Product[Product[f[k] - f[j], {j, 1, k-1}], {k, 2, n}];

%t d[n_]:= Product[(i - 1)!, {i, 1, n}];

%t Table[v[n], {n,z}] (* this sequence *)

%t Table[v[n+1]/v[n], {n,z}] (* A203419 *)

%t Table[v[n]/d[n], {n,z}] (* A203420 *)

%o (Magma)

%o A002808:=[n: n in [2..250] | not IsPrime(n)];

%o a:= func< n | n eq 0 select 1 else (&*[(&*[A002808[k+2] - A002808[j+1]: j in [0..k]]): k in [0..n-1]]) >;

%o [a(n): n in [0..20]]; // _G. C. Greubel_, Feb 24 2024

%o (SageMath)

%o A002808=[n for n in (2..250) if not is_prime(n)]

%o def a(n): return product(product( A002808[k+1] - A002808[j] for j in range(k+1)) for k in range(n))

%o [a(n) for n in range(15)] # _G. C. Greubel_, Feb 24 2024

%Y Cf. A000040, A000178, A002808, A203419, A203420.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2012