OFFSET
0,3
COMMENTS
Squarefree factorials: a(1) = 1, a(n+1) = a(n)* largest squarefree divisor of (n+1). - Amarnath Murthy, Nov 28 2004
LCM over all partitions of n of the product of the part sizes in the partition. - Franklin T. Adams-Watters, May 04 2010
a(n) is the product of the lcm of the set of prime factors of k over the range k = 1..n. - Peter Luschny, Jun 10 2011
a(n) is a divisor of n! and n!/a(n) = A085056(n). - Robert FERREOL, Aug 09 2021
In consequence of the definition, pseudo-binomial coefficients a(m+n)/(a(m)*a(n)) are natural numbers for all whole numbers m and n, and this is the minimal increasing sequence (for n >= 1) with that property. In consequence of the comment of Adams-Watters, the corresponding pseudo-multinomial coefficients are natural numbers as well. - Hal M. Switkay, May 26 2024
REFERENCES
Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued Polynomials, AMS, Providence, RI, 1997. Math. Rev. 98a:13002. See p. 246.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See rho_n p. 3.
Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018.
FORMULA
a(0) = 1, a(1) = 1; for n > 1, a(n) = lcm( 1, 2, ..., n, a(1)*a(n-1), a(2)*a(n-2), ..., a(n-1)*a(1) ). [Original name.]
a(n) = Product_{p prime} p^floor(n/p). See Farhi link p. 16. - Michel Marcus, Oct 18 2018
For n >=1, a(n) = lcm(1^floor(n/1),2^floor(n/2),...,n^floor(n/n)). - Robert FERREOL, Aug 05 2021
Rephrasing Murthy's comment: a(n) = a(n-1) * A007947(n). - Hal M. Switkay, Dec 31 2024
MAPLE
A048803 := proc(n) local i; mul(ilcm(op(numtheory[factorset](i))), i=1..n) end; seq(A048803(i), i=0..22); # Peter Luschny, Jun 10 2011
a := n -> mul(NumberTheory:-Radical(i), i=1..n): # Peter Luschny, Mar 14 2022
MATHEMATICA
a[0] = 1; a[n_] := a[n] = a[n-1] First @ Select[Reverse @ Divisors[n], SquareFreeQ, 1]; Array[a, 22, 0] (* Jean-François Alcover, May 04 2011 *)
A048803[n_] := Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A048803[n], {n, 0, 24}] (* Peter Luschny, Aug 18 2025 *)
PROG
(PARI) a(n)=local(f); f=n>=0; if(n>1, forprime(p=2, n, f*=p^(n\p))); f
(Haskell)
a048803 n = a048803_list !! n
a048803_list = scanl (*) 1 a007947_list
-- Reinhard Zumkeller, Jul 01 2013
(SageMath)
from functools import cache
@cache
def a_rec(n):
if n == 0: return 1
return radical(n) * a_rec(n - 1)
print([a_rec(n) for n in range(23)]) # Peter Luschny, Dec 12 2023
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Christian G. Bower, Apr 15 1999
EXTENSIONS
Entry improved by comments from Michael Somos, Nov 24 2001
New name based on a comment of Amarnath Murthy by Peter Luschny, Aug 18 2025
STATUS
approved
