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A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

%I A305945 #6 Jun 23 2018 09:19:07
%S A305945 16,16,12,11,21,12,17,14,16,17,14,13,16,18,13,14,10,10,21,7,19,13,15,
%T A305945 13,10,15,12,15,11,11,15,10,9,15,17,16,13,12,12,11,14,9,14,15,16,14,
%U A305945 13,14,15,24,14
%N A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.
%C A305945 a(0) = 16 is the number of terms in A008839 and in A195948, which includes the power 5^0 = 1.
%C A305945 These are the row lengths of A305925. 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 A305945 M. F. Hasler, <a href="/wiki/Zeroless_Powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.
%H A305945 T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.
%H A305945 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 A305945 (PARI) A305945(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(5^k))==n)
%o A305945 (PARI) A305945_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(5^k)),nMax)]++);a[^-1]}
%Y A305945 Cf. A030701 (= row 0 of A305925): k such that 5^k has no 0's; A195948: these powers 4^k.
%Y A305945 Cf. A020665: largest k such that n^k has no '0's.
%Y A305945 Cf. A063585 (= column 1 of A305925): least k such that 5^k has n digits '0' in base 10.
%Y A305945 Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).
%K A305945 nonn,base
%O A305945 0,1
%A A305945 _M. F. Hasler_, Jun 22 2018