A007377 Numbers k such that the decimal expansion of 2^k contains no 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• base
• nonn

Comments

It is an open problem of long standing to show that 86 is the last term.

A027870(a(n)) = A224782(a(n)) = 0. - Reinhard Zumkeller, Apr 30 2013

See A030700 for the analog for 3^k, which seems to end with k=68. - M. F. Hasler, Mar 07 2014

Checked up to k = 10^10. - David Radcliffe, Aug 21 2022

Examples

			Here is 2^86, conjecturally the largest power of 2 not containing a 0: 77371252455336267181195264. - _N. J. A. Sloane_, Feb 10 2023
		

References

• J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
• Eric Weisstein's World of Mathematics, Zero
• R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993

Crossrefs

Cf. A027870, A030700, A102483, A034293.

Some similar sequences are listed in A035064.

Cf. also A031142.

Programs

Haskell

import Data.List (elemIndices)
a007377 n = a007377_list !! (n-1)
a007377_list = elemIndices 0 a027870_list
-- Reinhard Zumkeller, Apr 30 2013

Magma

[ n: n in [0..50000] | not 0 in Intseq(2^n) ];  // Bruno Berselli, Jun 08 2011

Maple

remove(t -> has(convert(2^t,base,10),0),[$0..1000]); # Robert Israel, Dec 29 2015

Mathematica

Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]
Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]
Select[Range[0,100],DigitCount[2^#,10,0]==0&] (* Harvey P. Dale, Feb 06 2015 *)

PARI

for(n=0,99,if(vecmin(eval(Vec(Str(2^n)))),print1(n", "))) \\ Charles R Greathouse IV, Jun 30 2011

Perl

use bignum;
for(0..99) {
  if((1<<$_) =~ /^[1-9]+$/) {
    print "$_, "
  }
} # Charles R Greathouse IV, Jun 30 2011

Python

def ok(n): return '0' not in str(2**n)
print(list(filter(ok, range(10**4)))) # Michael S. Branicky, Aug 08 2021

Extensions

a(1) = 0 prepended by Reinhard Zumkeller, Apr 30 2013