A305927 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 7^k has n digits '0' (conjectured).

0, 1, 2, 3, 6, 7, 10, 11, 19, 35, 4, 5, 8, 12, 14, 15, 18, 27, 43, 47, 51, 9, 16, 17, 20, 24, 26, 28, 29, 34, 38, 52, 93, 13, 21, 22, 23, 30, 31, 36, 37, 42, 44, 46, 49, 58, 25, 32, 33, 50, 53, 54, 59, 66, 122, 55, 56, 57, 61, 62, 64, 67, 72, 73, 74, 39, 40, 48, 60, 71, 77, 79, 96, 108

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 (7, 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 considers "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?

Links

Table of n, a(n) for n=0..73.

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

Example

The table reads:
n \ k's
0 : 0, 1, 2, 3, 6, 7, 10, 11, 19, 35 (= A030703)
1 : 4, 5, 8, 12, 14, 15, 18, 27, 43, 47, 51
2 : 9, 16, 17, 20, 24, 26, 28, 29, 34, 38, 52, 93
3 : 13, 21, 22, 23, 30, 31, 36, 37, 42, 44, 46, 49, 58
4 : 25, 32, 33, 50, 53, 54, 59, 66, 122
5 : 55, 56, 57, 61, 62, 64, 67, 72, 73, 74
...
Column 0 is A063606: least k such that 7^k has n digits '0' in base 10.
Row lengths are 10, 11, 12, 13, 9, 10, 9, 7, 10, 14, 21, 10, 18, 7, 11, 11, 12, 15, 17, 10, ... (not in OEIS).
Last term of the rows are (35, 51, 93, 58, 122, 74, 108, 131, 118, 152, 195, 192, 236, 184, 247, 243, 254, 286, 325, 292, ...), not in OEIS.
The inverse permutation is (0, 1, 2, 3, 10, 11, 4, 5, 12, 21, 6, 7, 13, 33, 14, 15, 22, 23, 16, 8, 24, 34, 35, 36, 25, 46, 26, 17, 27, 28, 37, ...), not in OEIS.
Number of '0's in 7^n = row number of n: (0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 2, 1, 0, 2, 3, 3, 3, 2, 4, 2, 1, 2, 2, 3, 3, 4, 4, ...), not in OEIS.
Number of '0's in 7^n = row number of n: (0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 2, 1, 0, 2, 3, 3, 3, 2, 4, 2, 1, 2, 2, 3, 3, 4, 4, ...), not in OEIS.

Mathematica

mx = 1000; g[n_] := g[n] = DigitCount[7^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( A305927_row(n, M=50*(n+1))=select(k->#select(d->!d, digits(7^k))==n, [0..M]), [0..19])

Crossrefs

Cf. A030703, A063606.

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

Sequence in context: A032858 A181498 A030703 * A271585 A351715 A285259

Adjacent sequences: A305924 A305925 A305926 * A305928 A305929 A305930

Keyword

nonn,base,tabf

Author

M. F. Hasler, Jun 19 2018

Status

approved