A305928 -id:A305928 - cp's OEIS frontend

A030704 Numbers k such that the decimal expansion of 8^k contains no zeros (probably finite).

0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27

Offset: 1

Integers in A007377 / 3. - M. F. Hasler, Mar 07 2014

Eric Weisstein's World of Mathematics, Zero

Cf. A007377 (analog for 2^n), A030700 (for 3^n), A030701 (for 4^n), A008839 (for 5^n), A030702 (for 6^n), A030703 and A195908 (for 7^n), A030705 (for 9^n), A030706 and A195946 (for 11^n), A195944 and A195945 (for 13^n).
Cf. A195942, A195943.
This is row 0 of A305928.

[n: n in [0..500] | not 0 in Intseq(8^n)]; // Vincenzo Librandi, Mar 08 2014

Select[Range[0,30],DigitCount[8^#,10,0]==0&] (* Harvey P. Dale, Jul 13 2016 *)

select( is(n)=vecmin(digits(8^n)), [0..30]) \\ M. F. Hasler, Mar 07 2014

Several edits (offset 1, initial 0, title rephrased) by M. F. Hasler, Mar 07 2014

A305929 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 9^k has n digits '0' (conjectured).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34, 5, 8, 9, 10, 25, 26, 36, 11, 15, 16, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 48, 54, 68, 41, 45, 56, 33, 35, 37, 44, 49, 53, 58, 64, 65, 38, 39, 40, 43, 46, 51, 52, 59, 61, 67, 82, 83, 106, 42, 47, 62, 66, 69, 72, 73, 76, 84, 89, 144, 27, 50

Offset: 0

M. F. Hasler, Jun 19 2018

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 (9, 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?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34 (= A030705)
1 : 5, 8, 9, 10, 25, 26, 36
2 : 11, 15, 16, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 48, 54, 68
3 : 41, 45, 56
4 : 33, 35, 37, 44, 49, 53, 58, 64, 65
5 : 38, 39, 40, 43, 46, 51, 52, 59, 61, 67, 82, 83, 106
...
Column 0 is A063626: least k such that 9^k has n digits '0' in base 10.
Row lengths are 12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, ... (A305939).
Last element of the rows (largest exponent such that 9^k has exactly n digits 0) are (34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, ...), A306119.
Inverse permutation is (0, 1, 2, 3, 4, 12, 5, 6, 13, 14, 15, 19, 7, 8, 9, 20, 21, 10, 22, 23, 24, 25, 26, 27, 28, 16, 17, 73, 29, 30, 31, 32, ...), not in OEIS.

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

Cf. A030705, A063626.

Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305928 (analog for 8^k).

mx = 1000; g[n_] := g[n] = DigitCount[9^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( A305929_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(9^k))==n,[0..M]), [0..10])
print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort(concat(%),,1)[1..99]))) \\ to show row lengths, last terms and the inverse permutation

Row n consists of the integers in (row n of A305933 divided by 2).

A305938 Number of powers of 8 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

14, 11, 15, 11, 6, 12, 10, 7, 14, 21, 9, 9, 15, 8, 6, 10, 8, 13, 11, 13, 7, 10, 12, 8, 16, 10, 10, 10, 9, 14, 18, 11, 15, 12, 9, 9, 10, 17, 8, 12, 8, 12, 9, 8, 8, 12, 10, 17, 12, 6, 16

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 14 is the number of terms in A030704 and in A195946, which includes the power 7^0 = 1.

These are the row lengths of A305928. It remains an open problem to provide a proof that these rows are complete (as are all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. 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. A030704 (= row 0 of A305928): k such that 8^k has no 0's; A195946: these powers 8^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063616 (= column 1 of A305928): least k such that 8^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305939 (analog for 9^k).

A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(8^k))==n)

A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(8^k)),nMax)]++);a[^-1]}

A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

27, 43, 77, 61, 69, 119, 115, 158, 159, 168, 216, 232, 202, 198, 244, 270, 229, 274, 241, 273, 364, 283, 413, 298, 408, 341, 378, 431, 404, 403, 465, 483, 472, 454, 467, 508, 540, 575, 485, 576, 511, 623, 538, 515, 560, 655, 647, 661, 648, 639, 752

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030704: exponents of powers of 8 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, Mar 07 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. A063616: least k such that 8^k has n digits 0 in base 10.
Cf. A305938: number of k's such that 8^k has n digits 0.
Cf. A305928: row n lists exponents of 8^k with n digits 0.
Cf. A030704: { k | 8^k has no digit 0 } : row 0 of the above.
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.

A306118_vec(nMax,M=99*nMax+199,x=8,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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A030704 Numbers k such that the decimal expansion of 8^k contains no zeros (probably finite).

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A305929 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 9^k has n digits '0' (conjectured).

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A305938 Number of powers of 8 having exactly n digits '0' (in base 10), conjectured.

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A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

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