A030000 - OEIS

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

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OFFSET

0,1

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

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)

FORMULA

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

EXAMPLE

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

MATHEMATICA

Table[ i=0; While[ StringPosition[ ToString[ 2^i ], ToString[ n ] ]=={}, i++ ]; i, {n, 0, 80} ]

PROG

(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

(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

CROSSREFS

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