login
Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).
1

%I #11 May 08 2020 17:50:34

%S 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 143,1001,1001,1001,1001,1001,91,1547,221,221,221,4199,323,323,323,

%U 2261,2261,24871,24871,572033,572033,572033,81719,408595,24035,312455

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

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

%H G. C. Greubel, <a href="/A163644/b163644.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial.</a>

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

%p a := proc(n) local p; mul(p,p=select(isprime,{$1..n})

%p minus numtheory[factorset](n!/iquo(n,2)!^2)) end:

%t A034386[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];

%t sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f + 1, n - f]/f!];

%t 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 Cf. A056040, A034386, A163641, A056610.

%K nonn

%O 0,7

%A _Peter Luschny_, Aug 02 2009