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A305926 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 6^k has n digits '0' (conjectured).

%I A305926 #19 Mar 14 2025 15:52:08
%S A305926 0,1,2,3,4,5,6,7,8,12,17,24,29,44,10,11,14,15,18,22,28,40,42,59,9,16,
%T A305926 20,21,26,30,31,33,37,38,39,45,46,49,51,53,63,13,23,25,27,32,34,35,36,
%U A305926 47,48,54,61,72,73,76,82,19,52,60,64,65,70,71,83,91,93,98,43,50
%N A305926 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 6^k has n digits '0' (conjectured).
%C A305926 The set of (nonempty) rows forms a partition of the nonnegative integers.
%C A305926 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305926 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (6, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305926 It remains an open problem to provide a proof that the rows are complete, in the same way as each of the terms of A020665 is unproved.
%C A305926 We can also decide that the rows are to be truncated as soon as no term is found within a sufficiently large search limit. (For all of the displayed rows, there is no additional term up to many orders of magnitude beyond the last term.) That way the rows are well-defined, but it is no longer guaranteed to have a partition of the integers.
%C A305926 The author finds this sequence "nice", i.e., appealing (as well as, e.g., the variant A305933 for basis 3) in view of the idea of partitioning the integers in such an elementary yet highly nontrivial way, and the remarkable fact that the rows are just roughly one line long. Will this property remain for large n, or else, how will the row lengths evolve?
%H A305926 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers.</a>. OEIS Wiki, March 2014.
%e A305926 The table reads:
%e A305926 n \ k's
%e A305926 0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44 (= A030702)
%e A305926 1 : 10, 11, 14, 15, 18, 22, 28, 40, 42, 59
%e A305926 2 : 9, 16, 20, 21, 26, 30, 31, 33, 37, 38, 39, 45, 46, 49, 51, 53, 63
%e A305926 3 : 13, 23, 25, 27, 32, 34, 35, 36, 47, 48, 54, 61, 72, 73, 76, 82
%e A305926 4 : 19, 52, 60, 64, 65, 70, 71, 83, 91, 93, 98
%e A305926 5 : 43, 50, 55, 58, 62, 66, 67, 75, 77, 78, 101, 106, 129, 134
%e A305926 ...
%e A305926 Column 0 is A063596: least k such that 6^k has n digits '0' in base 10.
%e A305926 Row lengths are 14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, ... (A305946).
%e A305926 Last terms of the rows yield (44, 59, 63, 82, 98, 134, 108, 123, 199, 189, 192, 200, 275, 282, 267, 307, 298, 296, 391, 338, ...), A306116.
%e A305926 The inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 14, 15, 9, 41, 16, 17, 25, 10, 18, 57, 26, 27, 19, 42, 11, 43, 28, 44, 20, 12, 29, 30, ...), not in OEIS.
%t A305926 mx = 1000; g[n_] := g[n] = DigitCount[6^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* _Robert G. Wilson v_, Jun 20 2018 *)
%o A305926 (PARI) apply( A305926_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(6^k))==n,[0..M]), [0..19])
%Y A305926 Cf. A030702, A063596.
%Y A305926 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
%K A305926 nonn,base,tabf
%O A305926 0,3
%A A305926 _M. F. Hasler_, Jun 19 2018