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A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.

%I A305944 #6 Jun 23 2018 09:19:00
%S A305944 16,22,17,14,11,19,15,15,21,20,17,22,12,13,17,24,16,19,8,17,11,15,17,
%T A305944 15,20,17,18,20,17,21,16,19,16,14,15,19,20,24,7,16,13,14,13,14,22,22,
%U A305944 15,18,16,16,25
%N A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.
%C A305944 a(0) = 16 is the number of terms in A030701 and in A238940, which includes the power 4^0 = 1.
%C A305944 These are the row lengths of A305924. It remains an open problem to provide a proof that these rows are complete (as are 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.
%H A305944 M. F. Hasler, <a href="/wiki/Zeroless_Powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.
%H A305944 T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.
%H A305944 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 A305944 (PARI) A305944(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(4^k))==n)
%o A305944 (PARI) A305944_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(4^k)),nMax)]++);a[^-1]}
%Y A305944 Cf. A030701 = row 0 of A305924: k such that 4^k has no 0's; A238940: these powers 4^k.
%Y A305944 Cf. A020665: largest k such that n^k has no '0's.
%Y A305944 Cf. A063575 = column 1 of A305924: least k such that 4^k has n digits '0' in base 10.
%Y A305944 Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).
%K A305944 nonn,base
%O A305944 0,1
%A A305944 _M. F. Hasler_, Jun 22 2018