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Numbers k such that the decimal expansion of 2^k contains no 0.
(Formerly M0485)
56

%I M0485 #72 Feb 10 2023 16:37:52

%S 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 37,39,49,51,67,72,76,77,81,86

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

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

%C A027870(a(n)) = A224782(a(n)) = 0. - _Reinhard Zumkeller_, Apr 30 2013

%C See A030700 for the analog for 3^k, which seems to end with k=68. - _M. F. Hasler_, Mar 07 2014

%C Checked up to k = 10^10. - _David Radcliffe_, Aug 21 2022

%D J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H W. Schneider, <a href="/A007496/a007496.html">NoZeros: Powers n^k without Digit Zero</a> [Cached copy]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Zero.html">Zero</a>

%H R. G. Wilson, V, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>

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

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

%t Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]

%t Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]

%t Select[Range[0,100],DigitCount[2^#,10,0]==0&] (* _Harvey P. Dale_, Feb 06 2015 *)

%o (Magma) [ n: n in [0..50000] | not 0 in Intseq(2^n) ]; // _Bruno Berselli_, Jun 08 2011

%o (Perl) use bignum;

%o for(0..99) {

%o if((1<<$_) =~ /^[1-9]+$/) {

%o print "$_, "

%o }

%o } # _Charles R Greathouse IV_, Jun 30 2011

%o (PARI) for(n=0,99,if(vecmin(eval(Vec(Str(2^n)))),print1(n", "))) \\ _Charles R Greathouse IV_, Jun 30 2011

%o (Haskell)

%o import Data.List (elemIndices)

%o a007377 n = a007377_list !! (n-1)

%o a007377_list = elemIndices 0 a027870_list

%o -- _Reinhard Zumkeller_, Apr 30 2013

%o (Python)

%o def ok(n): return '0' not in str(2**n)

%o print(list(filter(ok, range(10**4)))) # _Michael S. Branicky_, Aug 08 2021

%Y Cf. A027870, A030700, A102483, A034293.

%Y Some similar sequences are listed in A035064.

%Y Cf. also A031142.

%K base,nonn

%O 1,3

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E a(1) = 0 prepended by _Reinhard Zumkeller_, Apr 30 2013