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

%I A203415 #10 Mar 02 2024 13:35:45
%S A203415 1,3,30,1680,201600,87091200,1103619686400,275463473725440000,
%T A203415 240529195987579699200000,1163776461866305616609280000000,
%U A203415 344605941225348705438623229542400000000,3717059729911125118574880410324812431360000000000
%N A203415 Vandermonde determinant of the first n nonprimes (A018252).
%C A203415 Each term divides its successor, as in A203416, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203417.
%H A203415 G. C. Greubel, <a href="/A203415/b203415.txt">Table of n, a(n) for n = 1..42</a>
%t A203415 z=20;
%t A203415 nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
%t A203415 f[j_]:= nonprime[[j]];
%t A203415 v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
%t A203415 d[n_]:= Product[(i-1)!, {i,1,n}];
%t A203415 Table[v[n], {n,1,z}]             (* this sequence *)
%t A203415 Table[v[n+1]/v[n], {n,1,z}]      (* A203416 *)
%t A203415 Table[v[n]/d[n], {n,1,z}]        (* A203417 *)
%o A203415 (Magma)
%o A203415 A018252:=[n : n in [1..250] | not IsPrime(n) ];
%o A203415 A203415:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;
%o A203415 [A203415(n): n in [1..30]]; // _G. C. Greubel_, Feb 29 2024
%o A203415 (SageMath)
%o A203415 A018252=[n for n in (1..250) if not is_prime(n)]
%o A203415 def A203415(n): return product(product(A018252[k+1]-A018252[j] for j in range(k+1)) for k in range(n-1))
%o A203415 [A203415(n) for n in range(1,31)] # _G. C. Greubel_, Feb 29 2024
%Y A203415 Cf. A000040, A018252, A203416, A203417.
%K A203415 nonn
%O A203415 1,2
%A A203415 _Clark Kimberling_, Jan 01 2012