A305945 -id:A305945 - cp's OEIS frontend

A305925 Irregular table read by rows in which row n >= 0 lists all k >= 0 such that the decimal representation of 5^k has n digits '0' (conjectured).

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58, 8, 12, 14, 15, 16, 19, 20, 21, 22, 25, 26, 31, 41, 42, 43, 85, 13, 23, 24, 27, 28, 29, 32, 36, 37, 56, 57, 107, 34, 35, 38, 39, 50, 54, 59, 74, 75, 84, 112, 40, 44, 46, 47, 49, 51, 60, 73, 78, 79, 82, 83, 86, 88, 89, 95, 96, 97, 106, 113, 127

Offset: 0

M. F. Hasler, Jun 19 2018

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 (5, 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 we are no more guaranteed to get 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?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58 (cf. A008839)
1 : 8, 12, 14, 15, 16, 19, 20, 21, 22, 25, 26, 31, 41, 42, 43, 85
2 : 13, 23, 24, 27, 28, 29, 32, 36, 37, 56, 57, 107
3 : 34, 35, 38, 39, 50, 54, 59, 74, 75, 84, 112
4 : 40, 44, 46, 47, 49, 51, 60, 73, 78, 79, 82, 83, 86, 88, 89, 95, 96, 97, 106, 113, 127
5 : 48, 55, 61, 67, 77, 91, 102, 110, 111, 126, 148, 157
...
The first column is A063585: least k such that 5^k has n digits '0' in base 10.
Row lengths are 16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7,... (A305945).
Last terms of the rows are (58, 85, 107, 112, 127, 157, 155, 194, 198, 238, 323, 237, 218, 301, 303, 324, 339, 476, 321, 284, ...), A306115.
The inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 16, 8, 9, 10, 17, 32, 18, 19, 20, 11, 12, 21, 22, 23, 24, 33, 34, 25, 26, 35, 36, 37, 13, 27,...), not in OEIS.

Cf. A008839, A063585.

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

mx = 1000; g[n_] := g[n] = DigitCount[5^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* Robert G. Wilson v, Jun 20 2018 *)

apply( A305925_row(n,M=60*(n+1))=select(k->#select(d->!d,digits(5^k))==n,[0..M]), [0..19])

A306115 Largest k such that 5^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

58, 85, 107, 112, 127, 157, 155, 194, 198, 238, 323, 237, 218, 301, 303, 324, 339, 476, 321, 284, 496, 421, 475, 415, 537, 447, 494, 538, 531, 439, 473, 546, 587, 588, 642, 690, 769, 689, 687, 686, 757, 732, 683, 826, 733, 825, 833, 810, 827, 888, 966

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A008839: exponents of powers of 5 without digit 0 in base 10.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A063585: least k such that 5^k has n digits 0 in base 10.
Cf. A305945: number of k's such that 5^k has n digits 0.
Cf. A305925: row n lists exponents of 5^k with n digits 0.
Cf. A008839: { k | 5^k has no digit 0 } : row 0 of the above.
Cf. A195948: { 5^k having no digit 0 }.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A071531: least k such that n^k contains a digit 0 in base 10.
Cf. A103663: least x such that x^n has no digit 0 in base 10.
Cf. A306112, ..., A306119: analog for 2^k, ..., 9^k.

A306115_vec(nMax,M=99*nMax+199,x=5,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

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A305925 Irregular table read by rows in which row n >= 0 lists all k >= 0 such that the decimal representation of 5^k has n digits '0' (conjectured).

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