A004643 -id:A004643 - 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

A004654 Powers of 2 written in base 15. (Next term contains a non-decimal character.)

Original entry on oeis.org

1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 918, 1331, 2662

Offset: 0

N. J. A. Sloane

a(14) is defined as "2^14 written in base 15". However, the base-15 digits of 2^14 are [4,12,12,4]. The standard convention of using letters A, B, ... to represent digits > 9, cannot be used in the Data sections of OEIS entries. One possibility to encode such terms within the given constraints and without affecting the earlier terms would be to use 00, 10, 20, ..., 50 to represent unambiguously the digits 0, 10, 11, ..., 14. - M. F. Hasler, Jun 22 2018.

This sequence makes a nice puzzle. It would be possible to allow non-decimal characters by using pairs of decimal digits instead of single digits, but this would spoil the beauty of the puzzle. - N. J. A. Sloane, Jun 25 2018

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

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

apply( a(n,b=15,m=2)=fromdigits(digits(m^n,b)), [0..13]) \\ This sums the base-15 digits multiplied by powers of 10. Digits > 9 occurring for n >= 14 will carry over to the left (4CC4 -> 5324). - M. F. Hasler, Jun 22 2018

A004667 Powers of 3 written in base 13. (Next term contains a non-decimal digit.)

Original entry on oeis.org

1, 3, 9, 21, 63, 159, 441

Offset: 0

N. J. A. Sloane

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.

FromDigits[IntegerDigits[#,13]]&/@(3^Range[0,6]) (* Harvey P. Dale, Mar 05 2018 *)

apply( a(n, b=13, m=3)=fromdigits(digits(m^n, b)), [0..6]) \\ This implements one possible continuation of the sequence beyond n = 6: write digits in decimal and carry over (so CC4 -> 12*100 + 12*10 + 4 = 1324). - M. F. Hasler, Jun 22 2018

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)

A372880 a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest proper multiple of a(n-1) that contains a(n-2) as subsequence of its digits.

Original entry on oeis.org

1, 3, 12, 36, 612, 1836, 168912, 10810368, 16366897152, 51703028103168, 1563447866811697152, 23520172003575940628103168, 1155558163424267804668132116971520, 12369352104691609178206055357839959406281031680

Offset: 1

Bryle Morga, May 15 2024

It is unknown whether a(n+1)/a(n) -> oo as n -> oo.

The same rule starting from terms 1, 2 gives A004643 and its multiples are as easy as A004643(n+1)/A004643(n) = 2 or 5 alternately.

			a(7) = 168912; 16812 = 92*1836 = 92*a(6) and "16812" contains a(5) = 612 as a subsequence.

Kevin Ryde, C Code

Cf. A004643.

C
```
/* See links. */
```

def subseq(x,y):
    i = 0
    j = 0
    while i != len(x) and j != len(y):
        if x[i] == y[j]:
           i += 1
        j += 1
    return i == len(x)
def a(n):
    if n == 1:
        return 1
    A = 1
    B = 3
    for _ in range(n-2):
        s = str(A)
        i = 1
        while not subseq(s, str(B*i)):
            i += 1
        A, B = B, B*i
    return B

from itertools import count, islice
def is_subseq(s, p):
    while s and p:
        if p%10 == s%10: s //= 10
        p //= 10
    return s == 0
def agen(): # generator of terms
    an2, an1 = [1, 3]
    yield from [an2, an1]
    while True:
        an = next(i*an1 for i in count(1) if is_subseq(an2, i*an1))
        an2, an1 = an1, an
        yield an
print(list(islice(agen(), 11))) # Michael S. Branicky, May 15 2024

a(n) <= a(n-2)*10^k + (a(n-1) - (a(n-2)*10^k mod a(n-1))), where k is the number of decimal digits in a(n-1). - Michael S. Branicky, May 17 2024

a(12)-a(13) from Michael S. Branicky, May 15 2024

a(14) from Kevin Ryde, Jun 23 2024

A305396 Records in A171797.

Original entry on oeis.org

101, 110, 211, 220, 321, 330, 431, 440, 541, 550

Offset: 1

N. J. A. Sloane, Jun 25 2018

The record-holders are the powers of 2 written in base 4, A004643.

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Cf. A171797, A004643.

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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A004654 Powers of 2 written in base 15. (Next term contains a non-decimal character.)

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A004667 Powers of 3 written in base 13. (Next term contains a non-decimal digit.)

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Mathematica

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A004669 Powers of 3 written in base 27.

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A372880 a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest proper multiple of a(n-1) that contains a(n-2) as subsequence of its digits.

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C

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A305396 Records in A171797.

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