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A296300 Numbers divisible by their length in every base. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A224470 A168247 A229056

Adjacent sequences:  A296297 A296298 A296299 * A296301 A296302 A296303

KEYWORD

nonn,base

AUTHOR

David Radcliffe, Dec 09 2017

STATUS

approved

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Last modified August 11 23:04 EDT 2020. Contains 336434 sequences. (Running on oeis4.)