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A163640 The radical of the swinging factorial A056040 for odd indices. 1
1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)$.

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

Table of n, a(n) for n=0..20.

Peter Luschny, Swinging Factorial.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Jul 30 2013 *)

CROSSREFS

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

Sequence in context: A056835 A056836 A277521 * A199130 A152743 A215906

Adjacent sequences:  A163637 A163638 A163639 * A163641 A163642 A163643

KEYWORD

nonn

AUTHOR

Peter Luschny, Aug 02 2009

STATUS

approved

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Last modified May 23 16:31 EDT 2017. Contains 286925 sequences.