A305924 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 4^k has n digits '0' (conjectured).

0, 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43, 5, 6, 10, 11, 13, 15, 19, 20, 22, 23, 24, 25, 29, 33, 34, 37, 42, 48, 61, 62, 65, 92, 21, 26, 27, 28, 30, 31, 32, 39, 40, 41, 46, 54, 58, 68, 74, 75, 77, 35, 45, 56, 57, 64, 66, 67, 70, 71, 78, 82, 83, 87, 88, 47, 53, 59, 63, 85, 89, 91, 93, 98

Offset: 0

Author

M. F. Hasler, Jun 14 2018

Keywords

• nonn
• base
• tabf

Comments

A partition of the nonnegative integers, the rows being the subsets.

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 (4, 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 "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?

Examples

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43 (= A030701)
1 : 5, 6, 10, 11, 13, 15, 19, 20, 22, 23, 24, 25, 29, 33, 34, 37, 42, 48, 61, 62, 65, 92
2 : 21, 26, 27, 28, 30, 31, 32, 39, 40, 41, 46, 54, 58, 68, 74, 75, 77
3 : 35, 45, 56, 57, 64, 66, 67, 70, 71, 78, 82, 83, 87, 88
4 : 47, 53, 59, 63, 85, 89, 91, 93, 98, 104, 115
5 : 44, 49, 52, 60, 72, 73, 76, 79, 80, 84, 90, 96, 109, 110, 114, 116, 120, 129, 171
...
Column 0 is A063575: least k such that 4^k has n digits '0' in base 10.
Row lengths are 16, 22, 17, 14, 11, ... = A305944.
Largest terms of the rows are 43, 92, 77, 88, 115, ... = A306114.
The inverse permutation is (0, 1, 2, 3, 4, 16, 17, 5, 6, 7, 18, 19, 8, 20, 9, 21, 10, 11, 12, 22, 23, 38, 24, 25, 26, 27, 39, 40, 41, 28, 42, 43, ...), not in OEIS.
		

Links

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

Crossrefs

Cf. A030701, A063575.

Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305925 (analog for 5^k), ..., A305929 (analog for 9^k).

Programs

Mathematica

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

PARI

apply( A305924_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(4^k))==n,[0..M]), [0..19])
print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort( concat(%),,1)[1..99]))) \\ to show row lengths, last terms & inverse permutation

Formula

Row n is given by the even terms of row n of A305932, divided by 2.