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%I #16 Apr 07 2020 22:27:23
%S 1024,2048,4096,131072,1048576,2097152,4194304,8388608,67108864,
%T 536870912,1073741824,274877906944,1099511627776,2199023255552,
%U 4398046511104,8796093022208,17592186044416,35184372088832,70368744177664,140737488355328,281474976710656,1125899906842624
%N Powers of 2 with the digit '0' in their decimal expansion.
%C The complement, A238938, is conjectured to be finite. Furthermore, Khovanova (see link) believes 2^86 = 77371252455336267181195264 is the largest power of 2 not in this sequence.
%H Tanya Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">"86 Conjecture"</a>, Tanya Khovanova's Math Blog, February 15, 2011.
%e 2^12 = 4096 contains one 0 in its decimal representation, hence 4096 is in the sequence.
%e 2^13 = 8192 contains no 0's and is thus not in the sequence.
%t Select[2^Range[0, 63], DigitCount[#, 10, 0] > 0 &]
%o (PARI) lista(nn) = {for (n=0, nn, if (vecsearch(Set(digits(p=2^n)), 0), print1(p, ", ")););} \\ _Michel Marcus_, Mar 05 2018
%Y Cf. A007377, A027870.
%K nonn,base
%O 1,1
%A _Alonso del Arte_, Jan 22 2018