A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

1, 1, 2, 3, 6, 10, 30, 105, 70, 42, 210, 2310, 2310, 4290, 6006, 15015, 30030, 170170, 510510, 1939938, 1385670, 881790, 9699690, 223092870, 44618574, 17160990, 74364290, 31870410, 223092870, 6469693230, 6469693230, 100280245065

Offset

0,3

Comments

Also squarefree kernel of A001142; row products in table A256113. - Reinhard Zumkeller, Mar 21 2015

a(2372) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017

Also the squarefree kernel of the cumulative product of n^n/n!. - Peter Luschny, Dec 21 2019

Conjecture: the few odd values belong to A070826. - Bill McEachen, Jun 24 2023

And their indices appear to be A007053. - Michel Marcus, Jul 01 2023

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2371

Formula

a(n) = A007947(A002944(n+1)). - Michel Marcus, Dec 21 2019

a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - Peter Luschny, Dec 21 2019

Example

a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
a(0) = 1 because n^n/n! = 1 for the integer n = 0. - Peter Luschny, Dec 21 2019

Maple

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
seq(rad(h(n)), n=0..31); # Peter Luschny, Dec 21 2019

Mathematica

Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* Michael De Vlieger, Jul 14 2017 *)

Prog

(Haskell)
a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015
(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = rad(lcm(vector(n+1, k, binomial(n, k-1)))); \\ Michel Marcus, Jun 24 2023

Crossrefs

Cf. A002944, A034386, A048633, A003418, A000142, A001405.

Cf. A007947, A001142, A256113, A002109, A000178, A329947.

Cf. A070826.

Sequence in context: A327711 A274964 A054357 * A371505 A186408 A062527

Adjacent sequences: A056603 A056604 A056605 * A056607 A056608 A056609

Keyword

nonn

Author

Labos Elemer, Aug 07 2000

Extensions

Extended with a(0) = 1 by Peter Luschny, Dec 21 2019

Status

approved