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A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

%I A306119 #8 Feb 11 2023 20:32:16
%S A306119 34,36,68,56,65,106,144,134,119,138,154,186,194,191,219,208,247,267,
%T A306119 199,314,292,263,319,303,307,345,431,401,375,388,413,498,488,504,465,
%U A306119 513,565,464,481,541,568,532,588,542,600,677,649,633,613,734,627
%N A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.
%C A306119 a(0) is the largest term in A030705: exponents of powers of 9 without digit 0 in base 10.
%C A306119 There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the Khovanova link.
%H A306119 M. F. Hasler, <a href="/wiki/Zeroless_Powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.
%H A306119 T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.
%H A306119 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 A306119 (PARI) A306119_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}
%Y A306119 Cf. A063626: least k such that 9^k has n digits 0 in base 10.
%Y A306119 Cf. A305939: number of k's such that 9^k has n digits 0.
%Y A306119 Cf. A305929: row n lists exponents of 9^k with n digits 0.
%Y A306119 Cf. A030705: { k | 9^k has no digit 0 } : row 0 of the above.
%Y A306119 Cf. A020665: largest k such that n^k has no digit 0 in base 10.
%Y A306119 Cf. A071531: least k such that n^k contains a digit 0 in base 10.
%Y A306119 Cf. A103663: least x such that x^n has no digit 0 in base 10.
%Y A306119 Cf. A306112, ..., A306118: analog for 2^k, ..., 8^k.
%K A306119 nonn,base
%O A306119 0,1
%A A306119 _M. F. Hasler_, Jun 22 2018
%E A306119 Data corrected thanks to a remark by _R. J. Mathar_, by _M. F. Hasler_, Feb 11 2023