A004642 -id:A004642 - cp's OEIS frontend

A364089 a(n) is the greatest k such that the base-n digits of 2^k are all distinct.

1, 1, 3, 4, 5, 8, 5, 10, 29, 19, 19, 19, 16, 18, 7, 43, 41, 37, 45, 39, 55, 33, 43, 60, 35, 61, 56, 50, 44, 69, 9, 64, 44, 80, 43, 88, 53, 71, 56, 68, 59, 78, 76, 74, 95, 109, 111, 81, 86, 136, 117, 75, 98, 83, 84, 99, 104, 116, 95, 118, 60, 81, 11, 119, 119, 172, 140, 97, 105, 113, 93, 122, 92

Offset: 2

Robert Israel, Jul 04 2023

a(n) <= log_2(A062813(n)).

			a(10) = 29 because all decimal digits of 2^29 = 536870912 are distinct.

Cf. A004642, A004643, A000866, A004645, A004646, A004647, A001357, A000079, A364049.

f:= proc(b) local M,k,L;
  M:= b^b - (b^b-b)/(b-1)^2;
  for k from ilog2(M) to 1 by -1 do
    L:= convert(2^k,base,b);
    if nops(L) = nops(convert(L,set)) then return k fi
  od
end proc:
map(f, [$2..100]);

from sympy.ntheory.factor_ import digits
def A364089(n):
    m = 1<<(l:=((r:=n**n)-(r-n)//(n-1)**2).bit_length()-1)
    while len(d:=digits(m,n)[1:]) > len(set(d)):
        l -= 1
        m >>= 1
    return l # Chai Wah Wu, Jul 07 2023

A004653 Powers of 2 written in base 14. (Next term contains a non-decimal character.)

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 92, 144, 288, 532

Offset: 0

N. J. A. Sloane

Next term contains a non-decimal character if such characters are chosen to represent digits > 9, where "digit" means the coefficients in N = Sum_{k>=0} d_k * b^k. This isn't possible here, but digits 0, 10, ..., 13 could be represented, e.g., using 00, 10, ..., 40. This would not affect a(0)..a(10), which don't have a digit 0. - M. F. Hasler, Jun 25 2018

Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 2, 3, ..., 16.

Cf. A000244, A004656, A004658, A004659, ...: powers of 3 in base 10, 2, 4, 5, ...

BaseForm[Table[2^n, {n, 0, 10}], 14] (* Alonso del Arte, Mar 18 2005 *)

apply( a(n)=fromdigits(digits(2^n,14)), [0..10]) \\ This yields Sum d[k]*10^k where d[k] are the base 14 digits. To get strings possibly containing letters 'A'..'D' replace fromdigits(...) by Strchr(apply(d->48+d+(d>9)*7,...)). - M. F. Hasler, Jun 25 2018

cp's OEIS Frontend

A364089 a(n) is the greatest k such that the base-n digits of 2^k are all distinct.

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