A368866 The smallest positive number such that 2^a(n) when written in base n contains adjacent equal digits.

2, 2, 4, 5, 6, 3, 6, 12, 16, 14, 11, 15, 8, 4, 8, 23, 16, 14, 16, 21, 9, 17, 20, 14, 30, 27, 16, 15, 10, 5, 10, 29, 48, 14, 46, 19, 18, 15, 32, 36, 27, 36, 18, 12, 56, 41, 37, 24, 58, 22, 26, 46, 58, 40, 29, 24, 36, 14, 20, 18, 12, 6, 12, 60, 62, 50, 49, 50, 20, 35, 36, 55, 61, 52, 53, 77

Offset: 2

Author

Scott R. Shannon, Jan 08 2024

Keywords

• nonn
• base

Comments

In the first 10000 terms the largest value is a(9031) = 1924, with a corresponding power of 2 of approximately 1.52*10^579.

Examples

			a(2) = 2 as 2^2 = 4 written in base 2 = 100_2 which contains adjacent 0's.
a(6) = 6 as 2^6 = 64 written in base 6 = 144_6 which contains adjacent 4's.
a(10) = 16 as 2^16 = 65536 written in base 10 = 65536_10 which contains adjacent 5's.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000079, A171901, A011557, A004642, A004643, A000866.

Programs

Maple

f:= proc(n) local k,L;
  for k from 1 do
    L:= convert(2^k,base,n);
    if member(0, L[2..-1]-L[1..-2]) then return k fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Jan 09 2024

Python

from itertools import count
from sympy.ntheory.factor_ import digits
def A368866(n):
    k = 1
    for m in count(1):
        k <<= 1
        s = digits(k,n)[1:]
        if any(s[i]==s[i+1] for i in range(len(s)-1)):
            return m # Chai Wah Wu, Jan 08 2024