A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).

1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 5, 35, 7, 7, 7, 21, 21, 105, 5, 55, 55, 165, 33, 429, 143, 1001, 1001, 1001, 1001, 1001, 91, 1547, 221, 221, 221, 4199, 323, 323, 323, 2261, 2261, 24871, 24871, 572033, 572033, 572033, 81719, 408595, 24035, 312455

Offset: 0

Author

Peter Luschny, Aug 02 2009

Keywords

• nonn

Examples

			a(20) = 105 because in the prime-factorization of 20$ the primes 3, 5 and 7 are missing and 3*5*7 = 105.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

Cf. A056040, A034386, A163641, A056610.

Programs

Maple

a := proc(n) local p; mul(p,p=select(isprime,{$1..n})
minus numtheory[factorset](n!/iquo(n,2)!^2)) end:

Mathematica

A034386[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f + 1, n - f]/f!];
A163641[0] = 1; A163641[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Join[{1}, Table[A034386[n]/A163641[n], {n, 1, 50}]] (* G. C. Greubel, Aug 01 2017 *)

Formula

a(n) = primorial(n) / rad(n$) = A034386(n) / A163641(n).