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A305924 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 4^k has n digits '0' (conjectured).

%I A305924 #29 Mar 14 2025 13:44:36
%S A305924 0,1,2,3,4,7,8,9,12,14,16,17,18,36,38,43,5,6,10,11,13,15,19,20,22,23,
%T A305924 24,25,29,33,34,37,42,48,61,62,65,92,21,26,27,28,30,31,32,39,40,41,46,
%U A305924 54,58,68,74,75,77,35,45,56,57,64,66,67,70,71,78,82,83,87,88,47,53,59,63,85,89,91,93,98
%N A305924 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 4^k has n digits '0' (conjectured).
%C A305924 A partition of the nonnegative integers, the rows being the subsets.
%C A305924 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305924 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (4, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305924 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 A305924 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 A305924 The author finds "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 A305924 M. F. Hasler, <a href="https://oeis.org/wiki/Zeroless_powers">Zeroless powers.</a>. OEIS Wiki, March 2014.
%F A305924 Row n is given by the even terms of row n of A305932, divided by 2.
%e A305924 The table reads:
%e A305924 n \ k's
%e A305924 0 : 0, 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43 (= A030701)
%e A305924 1 : 5, 6, 10, 11, 13, 15, 19, 20, 22, 23, 24, 25, 29, 33, 34, 37, 42, 48, 61, 62, 65, 92
%e A305924 2 : 21, 26, 27, 28, 30, 31, 32, 39, 40, 41, 46, 54, 58, 68, 74, 75, 77
%e A305924 3 : 35, 45, 56, 57, 64, 66, 67, 70, 71, 78, 82, 83, 87, 88
%e A305924 4 : 47, 53, 59, 63, 85, 89, 91, 93, 98, 104, 115
%e A305924 5 : 44, 49, 52, 60, 72, 73, 76, 79, 80, 84, 90, 96, 109, 110, 114, 116, 120, 129, 171
%e A305924 ...
%e A305924 Column 0 is A063575: least k such that 4^k has n digits '0' in base 10.
%e A305924 Row lengths are 16, 22, 17, 14, 11, ... = A305944.
%e A305924 Largest terms of the rows are 43, 92, 77, 88, 115, ... = A306114.
%e A305924 The inverse permutation is (0, 1, 2, 3, 4, 16, 17, 5, 6, 7, 18, 19, 8, 20, 9, 21, 10, 11, 12, 22, 23, 38, 24, 25, 26, 27, 39, 40, 41, 28, 42, 43, ...), not in OEIS.
%t A305924 mx = 1000; g[n_] := g[n] = DigitCount[4^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 A305924 (PARI) apply( A305924_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(4^k))==n,[0..M]), [0..19])
%o A305924 print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort( concat(%),,1)[1..99]))) \\ to show row lengths, last terms & inverse permutation
%Y A305924 Cf. A030701, A063575.
%Y A305924 Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305925 (analog for 5^k), ..., A305929 (analog for 9^k).
%K A305924 nonn,base,tabf
%O A305924 0,3
%A A305924 _M. F. Hasler_, Jun 14 2018