LINKS
G. C. Greubel, <a href="/A203417/b203417_1.txt">Table of n, a(n) for n = 1..135</a>
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G. C. Greubel, <a href="/A203417/b203417_1.txt">Table of n, a(n) for n = 1..135</a>
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Table[v[n]/d[n], {n, 1, z}] (* A203417 this sequence *)
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G. C. Greubel, <a href="/A203417/b203417_1.txt">Table of n, a(n) for n = 1..135</a>
z=20;
nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
f[j_]:= nonprime[[j]];
t v[n_]:= TableProduct[IfProduct[PrimeQf[k], 0, k - f[j], {j, 1, k, -1, 100}], {k, 2, n}];
nonprime = Rest[Union[t]];
f[j_] := nonprime[[j]]; z = 20; (* A018252 *)
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}] (* A203415 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A203416 *)
Table[v[n]/d[n], {n, 1, 20z}] (* A203417 *)
(Magma)
A018252:=[n : n in [1..250] | not IsPrime(n) ];
BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
v:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;
[v(n)/BarnesG(n+1): n in [1..30]]; // G. C. Greubel, Feb 29 2024
(SageMath)
A018252=[n for n in (1..250) if not is_prime(n)]
def BarnesG(n): return product(factorial(j) for j in range(1, n-1))
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))
[v(n)/BarnesG(n+1) for n in range(1, 31)] # G. C. Greubel, Feb 29 2024
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R. Chapman, <a href="httphttps://mathdlwww.maa.org/imagessites/default/cms_uploadfiles/Robin_Chapman27238.pdf">A polynomial taking integer values</a> , Mathematics Magazine 29 (1996), p. 121.
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_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 01 2012