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

%I A007377 M0485 #73 Feb 16 2025 08:32:31
%S A007377 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,
%T A007377 37,39,49,51,67,72,76,77,81,86
%N A007377 Numbers k such that the decimal expansion of 2^k contains no 0.
%C A007377 It is an open problem of long standing to show that 86 is the last term.
%C A007377 A027870(a(n)) = A224782(a(n)) = 0. - _Reinhard Zumkeller_, Apr 30 2013
%C A007377 See A030700 for the analog for 3^k, which seems to end with k=68. - _M. F. Hasler_, Mar 07 2014
%C A007377 Checked up to k = 10^10. - _David Radcliffe_, Aug 21 2022
%D A007377 J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.
%D A007377 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007377 W. Schneider, <a href="/A007496/a007496.html">NoZeros: Powers n^k without Digit Zero</a> [Cached copy]
%H A007377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zero.html">Zero</a>
%H A007377 R. G. Wilson, V, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>
%e A007377 Here is 2^86, conjecturally the largest power of 2 not containing a 0: 77371252455336267181195264. - _N. J. A. Sloane_, Feb 10 2023
%p A007377 remove(t -> has(convert(2^t,base,10),0),[$0..1000]); # _Robert Israel_, Dec 29 2015
%t A007377 Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]
%t A007377 Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]
%t A007377 Select[Range[0,100],DigitCount[2^#,10,0]==0&] (* _Harvey P. Dale_, Feb 06 2015 *)
%o A007377 (Magma) [ n: n in [0..50000] | not 0 in Intseq(2^n) ];  // _Bruno Berselli_, Jun 08 2011
%o A007377 (Perl) use bignum;
%o A007377 for(0..99) {
%o A007377   if((1<<$_) =~ /^[1-9]+$/) {
%o A007377     print "$_, "
%o A007377   }
%o A007377 } # _Charles R Greathouse IV_, Jun 30 2011
%o A007377 (PARI) for(n=0,99,if(vecmin(eval(Vec(Str(2^n)))),print1(n", "))) \\ _Charles R Greathouse IV_, Jun 30 2011
%o A007377 (Haskell)
%o A007377 import Data.List (elemIndices)
%o A007377 a007377 n = a007377_list !! (n-1)
%o A007377 a007377_list = elemIndices 0 a027870_list
%o A007377 -- _Reinhard Zumkeller_, Apr 30 2013
%o A007377 (Python)
%o A007377 def ok(n): return '0' not in str(2**n)
%o A007377 print(list(filter(ok, range(10**4)))) # _Michael S. Branicky_, Aug 08 2021
%Y A007377 Cf. A027870, A030700, A102483, A034293.
%Y A007377 Some similar sequences are listed in A035064.
%Y A007377 Cf. also A031142.
%K A007377 base,nonn
%O A007377 1,3
%A A007377 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007377 a(1) = 0 prepended by _Reinhard Zumkeller_, Apr 30 2013