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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). 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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?

LINKS

Table of n, a(n) for n=0..81.

EXAMPLE

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, ... (not in OEIS).

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, ...), not in OEIS.

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.

PROG

(PARI) 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.

CROSSREFS

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'.

Sequence in context: A122156 A325097 A030700 * A105208 A074779 A253567

Adjacent sequences:  A305930 A305931 A305932 * A305934 A305935 A305936

KEYWORD

nonn,base,tabf

AUTHOR

M. F. Hasler, Jun 14 2018

STATUS

approved

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Last modified October 21 14:29 EDT 2019. Contains 328301 sequences. (Running on oeis4.)