cp's OEIS Frontend

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

%I A305933 #28 Mar 14 2025 15:51:45
%S A305933 0,1,2,3,4,5,6,7,8,9,11,12,13,14,19,23,24,26,27,28,31,34,68,10,15,16,
%T A305933 17,18,20,25,29,43,47,50,52,63,72,73,22,30,32,33,36,38,39,40,41,42,44,
%U A305933 45,46,48,51,53,56,58,60,61,62,64,69,71,83,93,96,108,111,123,136,21,37,49,67,75,81,82,87,90,105,112,121,129
%N A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).
%C A305933 The set of nonempty rows is a partition of the nonnegative integers.
%C A305933 Read as a flattened sequence, a permutation of the nonnegative integers.
%C A305933 In the same way, another choice of (basis, digit, base) = (m, d, b) different from (3, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
%C A305933 It remains an open problem to provide a proof that the rows are complete, just as each of the terms of A020665 is unproved.
%C A305933 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 we are no longer guaranteed to get a partition of the integers.
%C A305933 The author finds the idea of partitioning the integers in this elementary yet highly nontrivial way appealing, as is the fact that the initial rows are just roughly one line long. Will this property continue to hold for large n, or if not, how will the row lengths evolve?
%e A305933 The table reads:
%e A305933 n \ k's
%e A305933 0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68 (cf. A030700)
%e A305933 1 : 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73
%e A305933 2 : 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136
%e A305933 3 : 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129
%e A305933 4 : 35, 59, 65, 66, 70, 74, 77, 79, 88, 98, 106, 116, 117, 128, 130, 131, 197, 205
%e A305933 5 : 57, 76, 78, 80, 86, 89, 91, 92, 101, 102, 104, 109, 115, 118, 122, 127, 134, 135, 164, 166, 203, 212, 237
%e A305933 ...
%e A305933 The first column is A063555: least k such that 3^k has n digits '0' in base 10.
%e A305933 Row lengths are 23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, ... (A305943).
%e A305933 Last term of the rows (i.e., largest k such that 3^k has exactly n digits 0) are (68, 73, 136, 129, 205, 237, 317, 268, 251, 276, 343, ...), A306113.
%e A305933 Inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 10, 11, 12, 13, 24, 25, 26, 27, 14, 28, 69, 38, 15, 16, 29, 17, 18, 19, 30, 39, 20, ...), not in OEIS.
%o A305933 (PARI) apply( A305933_row(n,M=50*n+70)=select(k->#select(d->!d,digits(3^k))==n,[0..M]), [0..10])
%o A305933 print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort(concat(%),,1)[1..99]))) \\ to show row lengths, last elements, and inverse permutation.
%Y A305933 Cf. A030700, A063555.
%Y A305933 Cf. A305932 (analog for 2^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
%Y A305933 Cf. A305934: powers of 3 with exactly one '0', A305943: powers of 3 with at least one '0'.
%K A305933 nonn,base,tabf
%O A305933 0,3
%A A305933 _M. F. Hasler_, Jun 14 2018