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A305924 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 4^k has n digits '0' (conjectured). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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?

LINKS

Table of n, a(n) for n=0..77.

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

FORMULA

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

EXAMPLE

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, 19, 15, 15, 21, 20, 17, 22, 12, 13, 17, 24, 16, 19, 8, 17, ... (not in OEIS).

Largest terms of the rows are (43, 92, 77, 88, 115, 171, 182, 238, 235, 308, 324, 348, 412, 317, 366, 445, 320, 424, 362, 448, ...), not in OEIS.

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.

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*)

PROG

(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

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).

Sequence in context: A064953 A097503 A030701 * A218785 A267373 A075061

Adjacent sequences:  A305921 A305922 A305923 * A305925 A305926 A305927

KEYWORD

nonn,base,tabf

AUTHOR

M. F. Hasler, Jun 14 2018

STATUS

approved

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Last modified August 20 20:50 EDT 2019. Contains 326155 sequences. (Running on oeis4.)