A030000 a(n) is the smallest nonnegative number k such that the decimal expansion of 2^k contains the string n.

10, 0, 1, 5, 2, 8, 4, 15, 3, 12, 10, 40, 7, 17, 18, 21, 4, 27, 30, 13, 11, 18, 43, 41, 10, 8, 18, 15, 7, 32, 22, 17, 5, 25, 27, 25, 16, 30, 14, 42, 12, 22, 19, 22, 18, 28, 42, 31, 11, 32, 52, 9, 19, 16, 25, 16, 8, 20, 33, 33, 23, 58, 18, 14, 6, 16, 46, 24, 15, 34, 29, 21, 17, 30

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

a(n) is well-defined for all n, because 2^k can actually start with (not just contain) any finite sequence of digits without leading zeros. This follows from the facts that log_10(2) is irrational and that the set of fractional parts of n*x is dense in [0,1] if x is irrational. - Pontus von Brömssen, Jul 21 2021

Examples

			2^12 = 4096 is first power of 2 containing a 9, so a(9) = 12.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..10000 (terms up to a(1000) from T. D. Noe, a(9634) corrected by Piotr Idzik)

Crossrefs

Cf. A030001 (the actual powers of 2), A063565, A018856, A000079, A372044, A372045.

See also A321043.

Programs

Haskell

import Data.List (isInfixOf, findIndex)
import Data.Maybe (fromJust)
a030000 n =
   fromJust $ findIndex (show n `isInfixOf`) $ map show a000079_list
-- Reinhard Zumkeller, Aug 04 2011

Mathematica

Table[ i=0; While[ StringPosition[ ToString[ 2^i ], ToString[ n ] ]=={}, i++ ]; i, {n, 0, 80} ]
snn[n_]:=Module[{k=0},While[SequenceCount[IntegerDigits[2^k],IntegerDigits[n]]==0,k++];k]; Array[snn,100,0] (* Harvey P. Dale, Mar 16 2025 *)

PARI

a(n) = {if (n==1, return (0)); my(k=1, sn = Str(n)); while (#strsplit(Str(2^k), sn) == 1, k++); k;} \\ Michel Marcus, Mar 06 2021

PARI

apply( A030000(n)={n=Str(n);for(k=0,oo,#strsplit(Str(2^k),n)>1&& return(k))}, [0..99]) \\ Also allows to search for digit strings with leading zeros, e.g., "00" => k=53. - M. F. Hasler, Jul 11 2021

Python

def a(n):
  k, strn = 0, str(n)
  while strn not in str(2**k): k += 1
  return k
print([a(n) for n in range(74)]) # Michael S. Branicky, Mar 06 2021

Formula

a(n) <= A018856(n) for n >= 1. - Pontus von Brömssen, Jul 21 2021

Extensions

More terms from Hans Havermann