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A203418 Vandermonde determinant of the first n composite numbers (A002808).

%I A203418 #12 Feb 24 2024 11:03:52
%S A203418 1,2,16,240,11520,13271040,254803968000,15892123484160000,
%T A203418 5126163351050649600000,89288743527804466888704000000,
%U A203418 50689719717698351557731837542400000000,125765178831579421305165126665125232640000000000
%N A203418 Vandermonde determinant of the first n composite numbers (A002808).
%C A203418 Each term divides its successor, as in A203419, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203420.
%H A203418 G. C. Greubel, <a href="/A203418/b203418.txt">Table of n, a(n) for n = 1..42</a>
%t A203418 composite = Select[Range[100], CompositeQ]; (* A002808 *)
%t A203418 z = 20;
%t A203418 f[j_]:= composite[[j]];
%t A203418 v[n_]:= Product[Product[f[k] - f[j], {j, 1, k-1}], {k, 2, n}];
%t A203418 d[n_]:= Product[(i - 1)!, {i, 1, n}];
%t A203418 Table[v[n], {n,z}]             (* this sequence *)
%t A203418 Table[v[n+1]/v[n], {n,z}]      (* A203419 *)
%t A203418 Table[v[n]/d[n], {n,z}]        (* A203420 *)
%o A203418 (Magma)
%o A203418 A002808:=[n: n in [2..250] | not IsPrime(n)];
%o A203418 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 A203418 [a(n): n in [0..20]]; // _G. C. Greubel_, Feb 24 2024
%o A203418 (SageMath)
%o A203418 A002808=[n for n in (2..250) if not is_prime(n)]
%o A203418 def a(n): return product(product( A002808[k+1] - A002808[j] for j in range(k+1)) for k in range(n))
%o A203418 [a(n) for n in range(15)] # _G. C. Greubel_, Feb 24 2024
%Y A203418 Cf. A000040, A000178, A002808, A203419, A203420.
%K A203418 nonn
%O A203418 1,2
%A A203418 _Clark Kimberling_, Jan 02 2012