

A094776


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


24




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

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 PC916.


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)
n = str(n)
for k in range(L if L else 10*20**len(n), 0, 1):


CROSSREFS

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


KEYWORD

nonn,fini,full,base


AUTHOR



STATUS

approved



