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

%I A163644 #16 Aug 22 2025 04:25:48
%S A163644 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,
%T A163644 143,1001,1001,1001,1001,1001,91,1547,221,221,221,4199,323,323,323,
%U A163644 2261,2261,24871,24871,572033,572033,572033,81719,408595,24035,312455
%N A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).
%H A163644 G. C. Greubel, <a href="/A163644/b163644.txt">Table of n, a(n) for n = 0..1000</a>
%H A163644 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163644 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163644 a(n) = primorial(n) / rad(n$) = A034386(n) / A163641(n).
%e A163644 a(20) = 105 because in the prime-factorization of 20$ the primes 3, 5 and 7 are missing and 3*5*7 = 105.
%p A163644 a := proc(n) local p; mul(p,p=select(isprime,{$1..n})
%p A163644 minus numtheory[factorset](n!/iquo(n,2)!^2)) end:
%t A163644 A034386[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];
%t A163644 sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f + 1, n - f]/f!];
%t A163644 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 *)
%Y A163644 Cf. A056040, A034386, A163641, A056610.
%K A163644 nonn
%O A163644 0,7
%A A163644 _Peter Luschny_, Aug 02 2009