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A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

%I A305942 #19 Apr 27 2020 02:42:59
%S A305942 36,41,31,34,25,32,37,23,43,47,33,35,29,27,27,39,34,34,28,29,31,30,38,
%T A305942 25,35,35,36,40,32,40,43,39,32,30,30,32,36,39,23,26,31,37,27,28,33,39,
%U A305942 28,44,34,27,43,33,27,32,31,27,27,32,35,34,36,28,32,39,38,40,28,43,38,32,22
%N A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.
%C A305942 a(0) = 36 is the number of terms in A007377 and in A238938, which includes the power 2^0 = 1.
%C A305942 These are the row lengths of A305932. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.
%C A305942 The average of the first 100000 terms is ~33.219 with a minimum of 12 and a maximum of 61. - _Hans Havermann_, Apr 26 2020
%H A305942 Hans Havermann, <a href="/A305942/b305942.txt">Table of n, a(n) for n = 0..10000</a>
%H A305942 M. F. Hasler, <a href="/wiki/Zeroless_Powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.
%H A305942 T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.
%H A305942 W. Schneider, <a href="http://web.archive.org/web/20050407120908/http://www.wschnei.de:80/digit-related-numbers/nozeros.html">No Zeros</a>, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)
%o A305942 (PARI) A305942(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(2^k))==n)
%o A305942 (PARI) A305942_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(2^k)),nMax)]++);a[^-1]}
%Y A305942 Row lengths of A305932 (row n = exponents of 2^k with n '0's).
%Y A305942 Cf. A007377 = {k | 2^k has no digit 0}; A238938: powers of 2 with no digit 0.
%Y A305942 Cf. A298607: powers of 2 with the digit '0' in their decimal expansion.
%Y A305942 Cf. A020665: largest k such that n^k has no digit 0 in base 10.
%Y A305942 Cf. A031146: least k such that 2^k has n digits 0 in base 10.
%Y A305942 Cf. A071531: least r such that n^r has a digit 0, in base 10.
%Y A305942 Cf. A306112: largest k such that 2^k has n digits 0, in base 10.
%K A305942 nonn,base
%O A305942 0,1
%A A305942 _M. F. Hasler_, Jun 21 2018