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A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

%I A094776 #26 Feb 13 2023 19:05:18
%S A094776 86,91,168,153,107,71,93,71,78,108
%N A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.
%C A094776 These values are only conjectural.
%C A094776 The sequence could be extended to any nonnegative integer index n defining a(n) to be the largest k such that n does not appear as substring in the decimal expansion of 2^k. I conjecture that for n = 10, 11, 12, ... it continues (2000, 3020, 1942, 1465, 1859, 2507, 1950, 1849, 1850, ...). For example, curiously enough, the largest power of 2 in which the string "10" does not appear seems to be 2^2000. - _M. F. Hasler_, Feb 10 2023
%D A094776 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.
%H A094776 Tanya Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">86 Conjecture</a>, T. K.'s Math Blog, Feb. 15, 2011.
%H A094776 Popular Computing (Calabasas, CA), <a href="/A094776/a094776.jpg">Two Tables</a>, Vol. 1, (No. 9, Dec 1973), page PC9-16.
%e A094776 a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.
%t A094776 f[n_] := Block[{a = {}, k = 1}, While[k < 10000, If[ Position[ Union[ IntegerDigits[ 2^k, 10]], n] == {}, AppendTo[a, k]]; k++ ]; a]; Table[ f[n][[ -1]], {n, 0, 9}] (* _Robert G. Wilson v_, Jun 12 2004 *)
%o A094776 (PARI) A094776(n,L=10*20^#Str(n))={forstep(k=L, 0, -1, foreach(digits(1<<k), d, d==n&&next(2)); return(k))} \\ _M. F. Hasler_, Feb 13 2023
%o A094776 (Python)
%o A094776 def A094776(n, L=0):
%o A094776    n = str(n)
%o A094776    for k in range(L if L else 10*20**len(n), 0, -1):
%o A094776       if n not in str(2**k): return k # _M. F. Hasler_, Feb 13 2023
%Y A094776 Cf. A259081 - A259083.
%Y A094776 Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).
%Y A094776 Cf. A034293 (numbers k such that 2^k has no '2').
%K A094776 nonn,fini,full,base
%O A094776 0,1
%A A094776 _Michael Taktikos_, Jun 09 2004