A075368 - OEIS

%I #16 Dec 05 2023 12:05:32

%S 1,6,10,5,84,84,1716,858,286,286,100776,100776,891480,891480,891480,

%T 445740,282861360,282861360,550835280,550835280,550835280,550835280,

%U 42222721680,42222721680,8444544336,8444544336,2814848112,2814848112

%N Smallest integer value of lcm(n+1, n+2, ..., n+k) (for k >= 0) divided by the product of all the primes up to n.

%H David A. Corneth, <a href="/A075368/b075368.txt">Table of n, a(n) for n = 1..2236</a>

%F a(n) = A075367(n)/A034386(n).

%e a(3) = 10 as the product of primes <= (n = 3) is 6 and the smallest integer of the form lcm(3+1, 3+2, ..., 3+k) = lcm(4, 5, 6) = 60 giving a(3) = 60/6 = 10. - _David A. Corneth_, Dec 05 2023

%t a75365[n_] := Module[{div, k, pr}, div=Times@@Prime/@Range[PrimePi[n]]; For[k=0; pr=1, True, k++; pr*=n+k, If[Mod[pr, div]==0, Return[k]]]]; a[1]=1; a[n_] := LCM@@Range[n+1, n+a75365[n]]/Times@@Prime/@Range[PrimePi[n]]

%o (PARI)

%o a(n) = {if(n==1, return(1));

%o 	my(pp = vecprod(primes(primepi(n))), l = n+1);

%o 	for(k = n+2, 2*n,

%o 		l = lcm(l, k);

%o 		if(l%pp == 0,

%o 			return(l\pp)

%o 		)

%o 	)

%o } \\ _David A. Corneth_, Dec 05 2023

%Y Cf. A075365, A075366, A075367.

%K nonn,easy

%O 1,2

%A _Amarnath Murthy_, Sep 20 2002

%E Edited by _Dean Hickerson_, Oct 28 2002