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A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

%I A305943 #18 Jun 25 2018 12:23:15
%S A305943 23,15,31,13,18,23,23,25,16,17,28,25,22,20,18,21,19,19,18,24,33,17,17,
%T A305943 18,17,14,21,26,25,23,24,29,17,22,18,21,27,26,20,21,13,27,24,12,18,24,
%U A305943 16,17,15,30,24,32,24,12,16,16,23,23,20,23,19,23,10,21,20,21,23,20,19,23,23,22,16,18,20,20,13,15,25,24,28,24,21,16,14,23,21,19,23,19,27,26,22,18,27,16,31,21,18,25,24
%N A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.
%C A305943 a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.
%C A305943 These are the row lengths of A305933. 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 vanishingly small, cf. Khovanova link.
%H A305943 M. F. Hasler, <a href="/wiki/Zeroless_powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.
%H A305943 T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.
%H A305943 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 A305943 (PARI) A305943(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(3^k))==n)
%o A305943 (PARI) A305943_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(3^k)),nMax)]++);a[^-1]}
%Y A305943 Cf. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.
%Y A305943 Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.
%Y A305943 Cf. A020665: largest k such that n^k has no '0's.
%Y A305943 Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.
%Y A305943 Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).
%K A305943 nonn,base
%O A305943 0,1
%A A305943 _M. F. Hasler_, Jun 22 2018