A296300 Numbers divisible by their length in every base.

1, 2, 6, 8, 12, 36, 48, 60, 240, 360, 540, 600, 660, 720, 840, 2100, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 8820, 10080, 11340, 12600, 13860, 15120, 16380, 17640, 20160, 21000, 21840, 22680, 23520, 24360

Offset

1,2

Comments

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

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

From Robert Israel, Dec 10 2017: (Start)

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

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

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

Maple

filter:= proc(n) local b, d;
  for d from 2 to ilog2(n)+1 do
    if n mod d <> 0 then
      b:= floor(n^(1/(d-1)));
      if b^d > n then return false fi;
    fi
  od;
  true
end proc:
select(filter, [$1..8, seq(i, i=12..10^6, 6)]); # Robert Israel, Dec 10 2017

Mathematica

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

Prog

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

Crossrefs

Cf. A003418.

Sequence in context: A113462 A065392 A030457 * A358970 A224470 A168247

Adjacent sequences: A296297 A296298 A296299 * A296301 A296302 A296303

Keyword

nonn,base

Author

David Radcliffe, Dec 09 2017

Status

approved