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A305927 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 7^k has n digits '0' (conjectured).

%I A305927 #24 Mar 14 2025 15:52:01
%S A305927 0,1,2,3,6,7,10,11,19,35,4,5,8,12,14,15,18,27,43,47,51,9,16,17,20,24,
%T A305927 26,28,29,34,38,52,93,13,21,22,23,30,31,36,37,42,44,46,49,58,25,32,33,
%U A305927 50,53,54,59,66,122,55,56,57,61,62,64,67,72,73,74,39,40,48,60,71,77,79,96,108
%N A305927 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 7^k has n digits '0' (conjectured).
%C A305927 The set of (nonempty) rows forms a partition of the nonnegative integers.
%C A305927 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305927 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (7, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305927 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 A305927 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 A305927 The author considers "nice", i.e., appealing, 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 A305927 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers.</a>. OEIS Wiki, March 2014.
%e A305927 The table reads:
%e A305927 n \ k's
%e A305927 0 : 0, 1, 2, 3, 6, 7, 10, 11, 19, 35 (= A030703)
%e A305927 1 : 4, 5, 8, 12, 14, 15, 18, 27, 43, 47, 51
%e A305927 2 : 9, 16, 17, 20, 24, 26, 28, 29, 34, 38, 52, 93
%e A305927 3 : 13, 21, 22, 23, 30, 31, 36, 37, 42, 44, 46, 49, 58
%e A305927 4 : 25, 32, 33, 50, 53, 54, 59, 66, 122
%e A305927 5 : 55, 56, 57, 61, 62, 64, 67, 72, 73, 74
%e A305927 ...
%e A305927 Column 0 is A063606: least k such that 7^k has n digits '0' in base 10.
%e A305927 Row lengths are 10, 11, 12, 13, 9, 10, 9, 7, 10, 14, 21, 10, 18, 7, 11, 11, 12, 15, 17, 10, ... (A305947).
%e A305927 Last term of the rows are (35, 51, 93, 58, 122, 74, 108, 131, 118, 152, 195, 192, 236, 184, 247, 243, 254, 286, 325, 292, ...), A306117.
%e A305927 The inverse permutation is (0, 1, 2, 3, 10, 11, 4, 5, 12, 21, 6, 7, 13, 33, 14, 15, 22, 23, 16, 8, 24, 34, 35, 36, 25, 46, 26, 17, 27, 28, 37, ...), not in OEIS.
%e A305927 Number of '0's in 7^n = row number of n: (0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 2, 1, 0, 2, 3, 3, 3, 2, 4, 2, 1, 2, 2, 3, 3, 4, 4, ...), not in OEIS.
%e A305927 Number of '0's in 7^n = row number of n: (0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 2, 1, 0, 2, 3, 3, 3, 2, 4, 2, 1, 2, 2, 3, 3, 4, 4, ...), not in OEIS.
%t A305927 mx = 1000; g[n_] := g[n] = DigitCount[7^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 A305927 (PARI) apply( A305927_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(7^k))==n,[0..M]), [0..19])
%Y A305927 Cf. A030703, A063606.
%Y A305927 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
%K A305927 nonn,base,tabf
%O A305927 0,3
%A A305927 _M. F. Hasler_, Jun 19 2018