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A305928 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 8^k has n digits '0' (conjectured).

%I A305928 #14 Nov 21 2018 02:34:46
%S A305928 0,1,2,3,5,6,8,9,11,12,13,17,24,27,4,7,10,15,16,19,22,25,28,32,43,14,
%T A305928 18,20,21,26,36,37,39,45,47,49,50,55,57,77,23,29,30,31,38,41,44,51,52,
%U A305928 58,61,42,53,59,62,65,69,33,40,48,56,60,64,73,76,80,86,114,119,35,46
%N A305928 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 8^k has n digits '0' (conjectured).
%C A305928 The set of (nonempty) rows forms a partition of the nonnegative integers.
%C A305928 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305928 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (8, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305928 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 A305928 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.
%H A305928 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers.</a>. OEIS Wiki, March 2014
%F A305928 Row n consists of integers in row n of A305932 divided by 3.
%e A305928 The table reads:
%e A305928 n \ k's
%e A305928 0 : 0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27 (= A030704)
%e A305928 1 : 4, 7, 10, 15, 16, 19, 22, 25, 28, 32, 43
%e A305928 2 : 14, 18, 20, 21, 26, 36, 37, 39, 45, 47, 49, 50, 55, 57, 77
%e A305928 3 : 23, 29, 30, 31, 38, 41, 44, 51, 52, 58, 61
%e A305928 4 : 42, 53, 59, 62, 65, 69
%e A305928 5 : 33, 40, 48, 56, 60, 64, 73, 76, 80, 86, 114, 119
%e A305928 ...
%e A305928 Column 0 is A063596: least k such that 8^k has n digits '0' in base 10.
%e A305928 Row lengths are 14, 11, 15, 11, 6, 12, 10, 7, 14, 21, 9, 9, 15, 8, 6, 10, 8, 13, ... (not in the OEIS).
%e A305928 The inverse permutation is (0, 1, 2, 3, 14, 4, 5, 15, 6, 7, 16, 8, 9, 10, 25, 17, 18, 11, 26, 19, 27, 28, 20, 40, 12, 21, 29, 13, 22, ...), also not in the OEIS.
%t A305928 mx = 1000; g[n_] := g[n] = DigitCount[8^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 A305928 (PARI) apply( A305928_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(8^k))==n,[0..M]), [0..7])
%Y A305928 Cf. A030704, A063596.
%Y A305928 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
%K A305928 nonn,base,tabf
%O A305928 0,3
%A A305928 _M. F. Hasler_, Jun 19 2018