A004647 -id:A004647 - cp's OEIS frontend

A364049 a(n) is the least k such that the base-n digits of 2^k are not all distinct.

2, 2, 4, 5, 6, 3, 6, 11, 16, 14, 11, 12, 8, 4, 8, 15, 16, 12, 16, 18, 9, 17, 15, 14, 24, 13, 16, 15, 10, 5, 10, 19, 24, 14, 21, 15, 18, 15, 19, 17, 17, 28, 18, 12, 24, 23, 31, 24, 31, 20, 26, 44, 35, 33, 25, 18, 36, 14, 14, 18, 12, 6, 12, 23, 45, 37, 38, 24, 20, 35, 36, 26, 51, 31, 33, 47, 34, 34

Offset: 2

Robert Israel, Jul 03 2023

			a(10) = 16 because 2^16 = 65536 does not have all distinct digits in base 10, while 2^k does have all distinct digits for 1 <= k <= 15.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A004642, A004643, A000866, A004645, A004646, A004647, A001357, A000079, A011557, A364052, A364089.

f:= proc(n) local k,L;
  for k from 2 do
    L:= convert(2^k,base,n);
    if nops(L) <> nops(convert(L,set)) then return k fi
  od;
end proc:
map(f, [$2..100]);

from itertools import count
from sympy.ntheory import digits
def a(n): return next(k for k in count(2) if len(set(d:=digits(1<Michael S. Branicky, Jul 05 2023

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

Original entry on oeis.org

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

A004669 Powers of 3 written in base 27.

Original entry on oeis.org

1, 3, 9, 10, 30, 90, 100, 300, 900, 1000, 3000, 9000, 10000, 30000, 90000, 100000, 300000, 900000, 1000000, 3000000, 9000000, 10000000, 30000000, 90000000, 100000000, 300000000, 900000000, 1000000000

Offset: 0

N. J. A. Sloane

Similar to powers of 2 in base 8 (A004647) or 16 (A004655). - M. F. Hasler, Jun 22 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,10).

Cf. A000079, A004643, ..., A004655: powers of 2 written in base 10, 4, 5, ..., 16.

Cf. A000244, A004656, A004658, A004659, ..., A004668: powers of 3 in base 10, 2, 4, 5, ..., 26.

Table[FromDigits[IntegerDigits[3^n, 27]], {n, 0, 100}] (* G. C. Greubel, Oct 12 2018 *)

apply( a(n)=3^(n%3)*10^(n\3), [0..20]) \\ M. F. Hasler, Jun 22 2018

a(n) = 3^(n mod 3)*10^floor(n/3). - M. F. Hasler, Jun 22 2018
From Chai Wah Wu, Sep 03 2020: (Start)
a(n) = 10*a(n-3) for n > 2.
G.f.: (-9*x^2 - 3*x - 1)/(10*x^3 - 1). (End)

A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

Original entry on oeis.org

0, 1, 8, 64, 1000, 8000, 64000, 1000000, 8000000, 64000000, 1000000000, 8000000000, 64000000000, 1000000000000, 8000000000000, 64000000000000, 1000000000000000, 8000000000000000, 64000000000000000, 1000000000000000000, 8000000000000000000, 64000000000000000000

Offset: 1

Antonio Roldán, Mar 30 2022

			64 is in the sequence because it is a perfect cube (64 = 4^3) whose digits appear in nonincreasing order.

Cf. A004647, A028820, A028864, A234848, A273045.

Intersection of A000578 and A009996.

Select[Range[0, 4*10^6]^3, Max@ Differences[IntegerDigits[#]] <= 0 &] (* Amiram Eldar, Mar 30 2022 *)

ok(n) = digits(n) == vecsort(digits(n),,4) && ispower(n,3)

a(n) = A004647(n-1)^3.

cp's OEIS Frontend

A364049 a(n) is the least k such that the base-n digits of 2^k are not all distinct.

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

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Examples

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Maple

Python

A004669 Powers of 3 written in base 27.

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Mathematica

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A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

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Mathematica

PARI

Formula