A163641 -id:A163641 - cp's OEIS frontend

A163640 The radical of the swinging factorial A056040 for odd indices.

1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270, 5786272150230, 526024740930, 1074920122770, 7524440859390, 25583098921926, 104300326374006, 1912172650190110

Offset: 0

Peter Luschny, Aug 02 2009

nonn

Let $ denote the swinging factorial. a(n) is the radical of (2*n+1)$ which is the product of the prime numbers dividing (2*n+1)$. It is the largest squarefree divisor of (2*n+1)$, and so also described as the squarefree kernel of (2*n+1)$.

			(2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).

a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Times @@ FactorInteger[sf[2*n + 1]][[All, 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 30 2013 *)

More terms from Michel Marcus, Aug 22 2025

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

Original entry on oeis.org

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

Peter Luschny, Aug 02 2009

nonn

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

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.

Cf. A056040, A034386, A163641, A056610.

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

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 *)

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

cp's OEIS Frontend

A163640 The radical of the swinging factorial A056040 for odd indices.

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