A306113 -id:A306113 - 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).

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, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73, 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, 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129

Offset: 0

M. F. Hasler, Jun 14 2018

The set of nonempty rows is a partition of the nonnegative integers.
Read as a flattened sequence, a permutation of the nonnegative integers.
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.
It remains an open problem to provide a proof that the rows are complete, just as each of the terms of A020665 is unproved.
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.
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?

			The table reads:
n \ k's
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)
1 : 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73
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
3 : 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129
4 : 35, 59, 65, 66, 70, 74, 77, 79, 88, 98, 106, 116, 117, 128, 130, 131, 197, 205
5 : 57, 76, 78, 80, 86, 89, 91, 92, 101, 102, 104, 109, 115, 118, 122, 127, 134, 135, 164, 166, 203, 212, 237
...
The first column is A063555: least k such that 3^k has n digits '0' in base 10.
Row lengths are 23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, ... (A305943).
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.
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.

Cf. A030700, A063555.
Cf. A305932 (analog for 2^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
Cf. A305934: powers of 3 with exactly one '0', A305943: powers of 3 with at least one '0'.

apply( A305933_row(n,M=50*n+70)=select(k->#select(d->!d,digits(3^k))==n,[0..M]), [0..10])
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.

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).

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