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

Author

Labos Elemer, Aug 07 2000

Keywords

• nonn

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

Examples

			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
		

Links

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

Crossrefs

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

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

Cf. A070826.

Programs

Haskell

a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015

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 *)

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

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

Extensions

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