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A305926 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 6^k has n digits '0' (conjectured). 3
0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, 10, 11, 14, 15, 18, 22, 28, 40, 42, 59, 9, 16, 20, 21, 26, 30, 31, 33, 37, 38, 39, 45, 46, 49, 51, 53, 63, 13, 23, 25, 27, 32, 34, 35, 36, 47, 48, 54, 61, 72, 73, 76, 82, 19, 52, 60, 64, 65, 70, 71, 83, 91, 93, 98, 43, 50 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The set of (nonempty) rows forms 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 (6, 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, in the same way 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 it is no longer guaranteed to have a partition of the integers.

The author finds this sequence "nice", i.e., appealing (as well as, e.g., the variant A305933 for basis 3) in view of 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?

LINKS

Table of n, a(n) for n=0..69.

M. F. Hasler, Zeroless powers.. OEIS Wiki, March 2014

EXAMPLE

The table reads:

n \ k's

0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44 (= A030702)

1 : 10, 11, 14, 15, 18, 22, 28, 40, 42, 59

2 : 9, 16, 20, 21, 26, 30, 31, 33, 37, 38, 39, 45, 46, 49, 51, 53, 63

3 : 13, 23, 25, 27, 32, 34, 35, 36, 47, 48, 54, 61, 72, 73, 76, 82

4 : 19, 52, 60, 64, 65, 70, 71, 83, 91, 93, 98

5 : 43, 50, 55, 58, 62, 66, 67, 75, 77, 78, 101, 106, 129, 134

...

Column 0 is A063596: least k such that 6^k has n digits '0' in base 10.

Row lengths are 14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, ... (not in OEIS).

Last terms of the rows yield (44, 59, 63, 82, 98, 134, 108, 123, 199, 189, 192, 200, 275, 282, 267, 307, 298, 296, 391, 338, ...), not in the OEIS.

The inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 14, 15, 9, 41, 16, 17, 25, 10, 18, 57, 26, 27, 19, 42, 11, 43, 28, 44, 20, 12, 29, 30, ...), not in OEIS.

MATHEMATICA

mx = 1000; g[n_] := g[n] = DigitCount[6^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* Robert G. Wilson v, Jun 20 2018 *)

PROG

(PARI) apply( A305926_row(n, M=50*(n+1))=select(k->#select(d->!d, digits(6^k))==n, [0..M]), [0..19])

CROSSREFS

Cf. A030702, A063596.

Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).

Sequence in context: A066418 A015845 A030702 * A033087 A084589 A271831

Adjacent sequences:  A305923 A305924 A305925 * A305927 A305928 A305929

KEYWORD

nonn,base,tabf

AUTHOR

M. F. Hasler, Jun 19 2018

STATUS

approved

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Last modified May 25 16:08 EDT 2019. Contains 323572 sequences. (Running on oeis4.)