A345022 - OEIS

%I #34 Jul 16 2021 06:45:36

%S 3,4,30,12,0,120,1260,840,0,2520,0,27720,0,0,6126120,720720,0,

%T 12252240,0,0,0,232792560,0,5354228880,0,26771144400,0,80313433200,0,

%U 4658179125600,72201776446800,0,0,0,0,144403552893600,0,0,0,5342931457063200,0

%N Smallest number divisible by all numbers 1 through n+1 except n, or 0 if impossible.

%C a(n) = 0 for all n in A024619.

%C a(n) > a(n+1) > 0 iff n > 2 and n is a power of 2 and n+1 is a prime or prime power. Does this occur only for n in {4, 8, 16, 256, 65536}? - _Jon E. Schoenfield_, Jun 07 2021

%C Probably yes. Those are the Fermat numbers minus 1 and the number 8 (which is the only power of 2 that is one less than a square number). - _J. Lowell_, Jun 08 2021

%H Jon E. Schoenfield, <a href="/A345022/b345022.txt">Table of n, a(n) for n = 2..2310</a>

%F a(n) = 0 if n divides f(n), f(n) otherwise, where f(n) = lcm(A003418(n-1), n+1). - _Jon E. Schoenfield_, Jun 05 2021

%e a(5)=12 because 12 is divisible by 1, 2, 3, 4, and 6; but not 5.

%e a(6)=0 because it's impossible for a number to be divisible by 1, 2, 3, 4, 5, and 7; but not 6. Any number divisible by both 2 and 3 is also divisible by 6.

%t Table[If[Mod[l=LCM@@Join[Range[n-1],{n+1}],n]==0,0,l],{n,2,50}] (* _Giorgos Kalogeropoulos_, Jun 25 2021 *)

%o (Magma) a:=[]; L:=1; for n in [2..42] do t:=Lcm(L,n+1); if t mod n eq 0 then a[n-1]:=0; else a[n-1]:=t; end if; L:=Lcm(L,n); end for; a; // _Jon E. Schoenfield_, Jun 05 2021

%o (Python) # generates initial segment of sequence

%o from math import gcd

%o from itertools import accumulate

%o def lcm(a, b): return a * b // gcd(a, b)

%o def aupton(nn):

%o     lcm1 = accumulate(range(1, nn), lcm)

%o     lcm2 = [lcm(k, n+1)  for n, k in enumerate(lcm1, start=2)]

%o     return [m*(m%n != 0) for n, m in enumerate(lcm2, start=2)]

%o print(aupton(42)) # _Michael S. Branicky_, Jun 25 2021

%Y Cf. A024619, A003418.

%K nonn

%O 2,1

%A _J. Lowell_, Jun 05 2021

%E a(16)-a(42) from _Jon E. Schoenfield_, Jun 05 2021