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M. F. Hasler, <a href="/wiki/Zeroless_Powerspowers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.

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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 vanishing, vanishingly small, cf. Khovanova link.

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

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 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

0,1

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

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 vanishing, cf. Khovanova link.

M. F. Hasler, <a href="/wiki/Zeroless_Powers">Zeroless powers</a>, OEIS Wiki, March 2014, updated 2018.

T. Khovanova, <a href="https://blog.tanyakhovanova.com/2011/02/86-conjecture/">The 86-conjecture</a>, Tanya Khovanova's Math Blog, Feb. 2011.

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.)

(PARI) A305943(n, M=99*n+199)=sum(k=0, M, #select(d->!d, digits(3^k))==n)

(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]}

Cf. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.

Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.

Cf. A020665: largest k such that n^k has no '0's.

Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.

Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

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M. F. Hasler, Jun 22 2018

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Powers of 3 having at least one digit '0' in their decimal representation.

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59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 10460353203, 31381059609, 847288609443, 68630377364883, 205891132094649, 1853020188851841, 5559060566555523, 50031545098999707, 150094635296999121, 450283905890997363, 1350851717672992089, 4052555153018976267, 12157665459056928801

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The analog of A298607 for 3^k instead of 2^k.

The complement A238939 is conjectured to have only 23 elements, the largest being 3^68. Thus, all larger powers of 3 are (conjectured to be) in this sequence. Each of the subsequences "powers of 3 with exactly k digits 0" is conjectured to be finite. Provided there is at least one such element for each k >= 0, this leads to a partition of the integers, given in A305933.

(PARI) for(n=0, 69, vecmin(digits(3^n))|| print1(3^n", "))

Cf. A030700 (= row 0 of A305933): decimal expansion of 3^n contains no zeros.

Complement of A238939: powers of 3 with no digit '0' in their decimal expansion.

Analog of A298607: powers of 2 with the digit '0' in their decimal expansion.

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