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

1, 3, 15, 140, 700, 2520, 44352, 2196480, 47567520, 634233600, 51753461760, 13984935444480, 1448751906201600, 82605199597240320, 32797812715211980800, 5296846753506734899200, 483765735240908144640000, 28985693293514522492928000

Offset: 1

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..135
• R. Chapman, A polynomial taking integer values, Mathematics Magazine 29 (1996), p. 121.
• Index to divisibility sequences

Crossrefs

Cf. A000040, A018252, A203415, A203416.

Programs

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

Mathematica

z=20;
nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
f[j_]:= nonprime[[j]];
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}]    (* A203416 *)
Table[v[n]/d[n], {n,1,z}]        (* this sequence *)

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