A364049 -id:A364049 - 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

A364052 a(n) is the least k such that no number with distinct base-n digits is the product of k (not necessarily distinct) primes.

Original entry on oeis.org

2, 3, 7, 9, 12, 14, 14, 21, 26, 28, 33, 36, 40, 45, 36, 50, 59, 61, 65, 70, 75, 77, 85, 89, 94, 97, 104, 107, 113, 118, 84

Offset: 2

Robert Israel, Jul 04 2023

A364049(n) <= a(n) <= 1 + floor(log_2(A062813(n))).

			a(4) = 7 because 2 = 2^1 = 2_4, 4 = 2^2 = 10_4, 8 = 2^3 = 20_4, 24 = 2^3 * 3 = 120_4, 108 = 2^2 * 3^3 = 1230_4 and 216 = 2^3 * 3^3 = 3120_4 have distinct base-4 digits and are products of 1 to 6 primes respectively, but there is no product of 7 primes that has distinct base-4 digits.

Cf. A001222, A062813, A165712, A364049.

f:= proc(n) local d,S,V,k;
  V:= {};
  for d from 1 to n do
    S:= select(t -> t[-1] <> 0, combinat:-permute([$0..n-1],d));
    S:= map(proc(t) local i;  numtheory:-bigomega(add(t[i]*n^(i-1),i=1..d)) end proc, S);
    V:= V union convert(S,set);
  od;
  min({$1..1+max(V)} minus V)
end proc:
map(f, [$2..10]);

a(11) from Jon E. Schoenfield, Jul 05 2023
a(12) from Martin Ehrenstein, Jul 07 2023
a(13)-a(18) from Jon E. Schoenfield, Jul 08 2023
a(19)-a(22) from Pontus von Brömssen, Jul 13 2023
a(23)-a(32) from Bert Dobbelaere, Jul 20 2023

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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