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A203417 a(n) = A203415(n)/A000178(n).

%I A203417 #21 May 27 2025 11:11:28
%S A203417 1,3,15,140,700,2520,44352,2196480,47567520,634233600,51753461760,
%T A203417 13984935444480,1448751906201600,82605199597240320,
%U A203417 32797812715211980800,5296846753506734899200,483765735240908144640000,28985693293514522492928000
%N A203417 a(n) = A203415(n)/A000178(n).
%H A203417 G. C. Greubel, <a href="/A203417/b203417.txt">Table of n, a(n) for n = 1..135</a>
%H A203417 R. Chapman, <a href="https://web.archive.org/web/20230227162446/https://www.maa.org/sites/default/files/Robin_Chapman27238.pdf">A polynomial taking integer values</a>, Mathematics Magazine 29 (1996), p. 121.
%H A203417 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%t A203417 z=20;
%t A203417 nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
%t A203417 f[j_]:= nonprime[[j]];
%t A203417 v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
%t A203417 d[n_]:= Product[(i-1)!, {i,1,n}];
%t A203417 Table[v[n], {n,1,z}]             (* A203415 *)
%t A203417 Table[v[n + 1]/v[n], {n,1,z}]    (* A203416 *)
%t A203417 Table[v[n]/d[n], {n,1,z}]        (* this sequence *)
%o A203417 (Magma)
%o A203417 A018252:=[n : n in [1..250] | not IsPrime(n) ];
%o A203417 BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203417 v:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;
%o A203417 [v(n)/BarnesG(n+1): n in [1..30]]; // _G. C. Greubel_, Feb 29 2024
%o A203417 (SageMath)
%o A203417 A018252=[n for n in (1..250) if not is_prime(n)]
%o A203417 def BarnesG(n): return product(factorial(j) for j in range(1, n-1))
%o A203417 def v(n): return product(product(A018252[k-1]-A018252[j-1] for j in range(1,k)) for k in range(2,n+1))
%o A203417 [v(n)/BarnesG(n+1) for n in range(1,31)] # _G. C. Greubel_, Feb 29 2024
%Y A203417 Cf. A000040, A018252, A203415, A203416.
%K A203417 nonn
%O A203417 1,2
%A A203417 _Clark Kimberling_, Jan 01 2012