cp's OEIS Frontend

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

%I A048803 #59 Aug 19 2025 02:25:51
%S A048803 1,1,2,6,12,60,360,2520,5040,15120,151200,1663200,9979200,129729600,
%T A048803 1816214400,27243216000,54486432000,926269344000,5557616064000,
%U A048803 105594705216000,1055947052160000,22174888095360000,487847538097920000,11220493376252160000,67322960257512960000
%N A048803 a(n) = Product_{k=1..n} rad(k), where rad(n) is the product of distinct prime factors of n, cf. A007947.
%C A048803 Squarefree factorials: a(1) = 1, a(n+1) = a(n)* largest squarefree divisor of (n+1). - _Amarnath Murthy_, Nov 28 2004
%C A048803 LCM over all partitions of n of the product of the part sizes in the partition. - _Franklin T. Adams-Watters_, May 04 2010
%C A048803 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
%C A048803 a(n) is a divisor of n! and n!/a(n) = A085056(n). - _Robert FERREOL_, Aug 09 2021
%C A048803 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
%D A048803 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued Polynomials, AMS, Providence, RI, 1997. Math. Rev. 98a:13002. See p. 246.
%H A048803 Reinhard Zumkeller, <a href="/A048803/b048803.txt">Table of n, a(n) for n = 0..500</a>
%H A048803 Abdelmalek Bedhouche and Bakir Farhi, <a href="https://arxiv.org/abs/2207.07957">On some products taken over the prime numbers</a>, arXiv:2207.07957 [math.NT], 2022. See rho_n p. 3.
%H A048803 Bakir Farhi, <a href="https://arxiv.org/abs/1810.07560">On the derivatives of the integer-valued polynomials</a>, arXiv:1810.07560 [math.NT], 2018.
%H A048803 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>
%F A048803 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.]
%F A048803 a(n) = Product_{p prime} p^floor(n/p). See Farhi link p. 16. - _Michel Marcus_, Oct 18 2018
%F A048803 For n >=1, a(n) = lcm(1^floor(n/1),2^floor(n/2),...,n^floor(n/n)). - _Robert FERREOL_, Aug 05 2021
%F A048803 Rephrasing Murthy's comment: a(n) = a(n-1) * A007947(n). - _Hal M. Switkay_, Dec 31 2024
%p A048803 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
%p A048803 a := n -> mul(NumberTheory:-Radical(i), i=1..n): # _Peter Luschny_, Mar 14 2022
%t A048803 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 *)
%t A048803 A048803[n_] := Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
%t A048803 Table[A048803[n], {n, 0, 24}]  (* _Peter Luschny_, Aug 18 2025 *)
%o A048803 (PARI) a(n)=local(f); f=n>=0; if(n>1, forprime(p=2,n,f*=p^(n\p))); f
%o A048803 (Haskell)
%o A048803 a048803 n = a048803_list !! n
%o A048803 a048803_list = scanl (*) 1 a007947_list
%o A048803 -- _Reinhard Zumkeller_, Jul 01 2013
%o A048803 (SageMath)
%o A048803 from functools import cache
%o A048803 @cache
%o A048803 def a_rec(n):
%o A048803     if n == 0: return 1
%o A048803     return radical(n) * a_rec(n - 1)
%o A048803 print([a_rec(n) for n in range(23)]) # _Peter Luschny_, Dec 12 2023
%Y A048803 Partial products of A007947.
%K A048803 nonn,nice,changed
%O A048803 0,3
%A A048803 _Christian G. Bower_, Apr 15 1999
%E A048803 Entry improved by comments from _Michael Somos_, Nov 24 2001
%E A048803 New name based on a comment of _Amarnath Murthy_ by _Peter Luschny_, Aug 18 2025