A048803 a(n) = Product_{k=1..n} rad(k), where rad(n) is the product of distinct prime factors of n, cf. A007947.

1, 1, 2, 6, 12, 60, 360, 2520, 5040, 15120, 151200, 1663200, 9979200, 129729600, 1816214400, 27243216000, 54486432000, 926269344000, 5557616064000, 105594705216000, 1055947052160000, 22174888095360000, 487847538097920000, 11220493376252160000, 67322960257512960000

Offset: 0

Author

Christian G. Bower, Apr 15 1999

Keywords

• nonn
• nice
• changed

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.
• Index entries for sequences related to lcm's

Crossrefs

Partial products of A007947.

Programs

Haskell

a048803 n = a048803_list !! n
a048803_list = scanl (*) 1 a007947_list
-- Reinhard Zumkeller, Jul 01 2013

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 *)

PARI

a(n)=local(f); f=n>=0; if(n>1, forprime(p=2,n,f*=p^(n\p))); f

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

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

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