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A305929 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 9^k has n digits '0' (conjectured).

%I A305929 #22 Mar 14 2025 15:52:05
%S A305929 0,1,2,3,4,6,7,12,13,14,17,34,5,8,9,10,25,26,36,11,15,16,18,19,20,21,
%T A305929 22,23,24,28,29,30,31,32,48,54,68,41,45,56,33,35,37,44,49,53,58,64,65,
%U A305929 38,39,40,43,46,51,52,59,61,67,82,83,106,42,47,62,66,69,72,73,76,84,89,144,27,50
%N A305929 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 9^k has n digits '0' (conjectured).
%C A305929 The set of (nonempty) rows forms a partition of the nonnegative integers.
%C A305929 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305929 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (9, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305929 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 A305929 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 A305929 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 A305929 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers.</a>. OEIS Wiki, March 2014.
%F A305929 Row n consists of the integers in (row n of A305933 divided by 2).
%e A305929 The table reads:
%e A305929 n \ k's
%e A305929 0 : 0, 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34 (= A030705)
%e A305929 1 : 5, 8, 9, 10, 25, 26, 36
%e A305929 2 : 11, 15, 16, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 48, 54, 68
%e A305929 3 : 41, 45, 56
%e A305929 4 : 33, 35, 37, 44, 49, 53, 58, 64, 65
%e A305929 5 : 38, 39, 40, 43, 46, 51, 52, 59, 61, 67, 82, 83, 106
%e A305929 ...
%e A305929 Column 0 is A063626: least k such that 9^k has n digits '0' in base 10.
%e A305929 Row lengths are 12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, ... (A305939).
%e A305929 Last element of the rows (largest exponent such that 9^k has exactly n digits 0) are (34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, ...), A306119.
%e A305929 Inverse permutation is (0, 1, 2, 3, 4, 12, 5, 6, 13, 14, 15, 19, 7, 8, 9, 20, 21, 10, 22, 23, 24, 25, 26, 27, 28, 16, 17, 73, 29, 30, 31, 32, ...), not in OEIS.
%t A305929 mx = 1000; g[n_] := g[n] = DigitCount[9^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 A305929 (PARI) apply( A305929_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(9^k))==n,[0..M]), [0..10])
%o A305929 print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort(concat(%),,1)[1..99]))) \\ to show row lengths, last terms and the inverse permutation
%Y A305929 Cf. A030705, A063626.
%Y A305929 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305928 (analog for 8^k).
%K A305929 nonn,base,tabf
%O A305929 0,3
%A A305929 _M. F. Hasler_, Jun 19 2018