A188666 Largest m <= n such that lcm(m, m+1, ..., n) = lcm(1, 2, ..., n).

1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 7, 7, 8, 8, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 32, 32, 37, 37, 37, 37, 37, 37

Offset: 1

Author

Reinhard Zumkeller, Apr 25 2011

Keywords

• nonn

Comments

By definition: A003418(n) = lcm(a(n), a(n)+1, ... n)

and lcm(m, m+1, ... n) < A003418(n) for m > a(n);

all terms are prime powers, cf. A000961: A010055(a(n)) = 1;

a(A110654(n)) = A000015(n);

floor(n/2)+1 <= a(n) < a(2*(a(n));

A000961(n+1) = a(2*A000961(n)) = a(A138929(n)), cf. formula.

A237709 gives number of occurrences of n-th prime power. - Reinhard Zumkeller, Feb 12 2014

Conjecture: a(n) = A000015(floor(n/2)+1). - Georg Fischer, Nov 29 2022

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Least Common Multiple
• Wikipedia, Least Common Multiple
• Index entries for sequences related to lcm's

Crossrefs

Cf. A000015, A051173, A099996.

Programs

Haskell

import Data.List (elemIndices)
a188666 n = a188666_list !! (n-1)
a188666_list = g 1 a000961_list where
   g n pps'@(pp:pp':pps) | n < 2*pp  = pp  : g (n+1) pps'
                         | otherwise = pp' : g (n+1) (pp':pps)
-- Alternative, rewriting the definition, but less efficient:
a188666' n = last $ elemIndices (f 1) $ map f [0..n] where
   f from = foldl lcm 1 [from..n]

Mathematica

Table[Block[{k = n, m = LCM @@ Range[n]},  While[LCM @@ Range[k, n] != m, k--]; k], {n, 69}] (* Michael De Vlieger, Nov 29 2022 *)

PARI

A188666(n)=L=lcm(n=vector(n-1,k,k+1));!for(m=1,#n,lcm(n[-m..-1])==L&&return(#n+2-m))\\ Rather illustrative than efficient. - M. F. Hasler, Jul 25 2015

Python

from itertools import count
from sympy import factorint
def A188666(n): return next(filter(lambda m:len(factorint(m))<=1, count((n>>1)+1))) # Chai Wah Wu, Oct 25 2024

Formula

a(n) = A000961(k+1) for n: 2*A000961(k) <= n < 2*A000961(k+1), k > 0.