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A296300 Numbers divisible by their length in every base. 1

%I

%S 1,2,6,8,12,36,48,60,240,360,540,600,660,720,840,2100,2280,2400,2520,

%T 2640,2760,2880,3000,3120,3240,3360,3480,3600,3720,3840,3960,4080,

%U 8820,10080,11340,12600,13860,15120,16380,17640,20160,21000,21840,22680,23520,24360

%N Numbers divisible by their length in every base.

%C The infinitude of this sequence follows from the fact that lcm(1,2,...,n) = exp(n(1+o(1))).

%C For each positive integer n, all but finitely many terms are divisible by n.

%C From _Robert Israel_, Dec 10 2017: (Start)

%C Numbers n such that for every m>=2 not dividing n, floor(n^(1/(m-1))) <= n^(1/m).

%C All terms > 1 are even, all terms > 8 are divisible by 12. (End)

%C The number of terms < 10^k: 4, 8, 15, 33, 72, 134, 859, 1123, ..., . - _Robert G. Wilson v_, Dec 11 2017

%H Robert Israel, <a href="/A296300/b296300.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence contains 8 because the base-b representation of 8 has 1 digit when b > 8, 2 digits when 2 < b <= 8, and 4 digits when b = 2. In each case, the number of digits is a divisor of 8.

%p filter:= proc(n) local b,d;

%p for d from 2 to ilog2(n)+1 do

%p if n mod d <> 0 then

%p b:= floor(n^(1/(d-1)));

%p if b^d > n then return false fi;

%p fi

%p od;

%p true

%p end proc:

%p select(filter, [$1..8, seq(i,i=12..10^6,6)]); # _Robert Israel_, Dec 10 2017

%t {1}~Join~Select[Range[2, 10^5], Function[b, AllTrue[Range[2, b], Divisible[b, IntegerLength[b, #]] &]]] (* _Michael De Vlieger_, Dec 09 2017 *)

%t fQ[n_] := Block[{b = 2, lmt = Floor[ Sqrt[n +1] +2]}, While[b < lmt, If[ Mod[n, Ceiling[ Log[b, n]]] > 0, b = 0; Break[]]; b++]; b > 0]; Join[{1, 2, 6, 8, 12, 36, 48}, Select[60 Range@425, fQ]] (* _Robert G. Wilson v_, Dec 11 2017 *)

%o (Python) [n for n in range(1, 100000) if all(n%k == 0 or n**(1/k) >= int(n**(1/(k-1))) for k in range(2, len(bin(n))-1))]

%o (PARI) isok(n) = {for (b=2, n, if (n % #digits(n, b), return (0));); return (1);} \\ _Michel Marcus_, Dec 10 2017

%Y Cf. A003418.

%K nonn,base

%O 1,2

%A _David Radcliffe_, Dec 09 2017

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Last modified September 30 05:03 EDT 2020. Contains 337435 sequences. (Running on oeis4.)