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

86, 91, 168, 153, 107, 71, 93, 71, 78, 108

Offset

0,1

Comments

These values are only conjectural.

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

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Links

Table of n, a(n) for n=0..9.

Tanya Khovanova, 86 Conjecture, T. K.'s Math Blog, Feb. 15, 2011.

Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Example

a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

Mathematica

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 *)

Prog

(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
(Python)
def A094776(n, L=0):
   n = str(n)
   for k in range(L if L else 10*20**len(n), 0, -1):
      if n not in str(2**k): return k # M. F. Hasler, Feb 13 2023

Crossrefs

Cf. A259081 - A259083.

Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).

Cf. A034293 (numbers k such that 2^k has no '2').

Sequence in context: A058907 A045101 A020215 * A095595 A095581 A097399

Adjacent sequences: A094773 A094774 A094775 * A094777 A094778 A094779

Keyword

nonn,fini,full,base

Author

Michael Taktikos, Jun 09 2004

Status

approved