A063626 -id:A063626 - cp's OEIS frontend

A031146 Exponent of the least power of 2 having exactly n zeros in its decimal representation.

0, 10, 42, 43, 79, 88, 100, 102, 189, 198, 242, 250, 252, 263, 305, 262, 370, 306, 368, 383, 447, 464, 496, 672, 466, 557, 630, 629, 628, 654, 657, 746, 771, 798, 908, 913, 917, 906, 905, 1012, 1113, 988, 1020, 989, 1044, 1114, 1120, 1118, 1221, 1218, 1255

Offset: 0

N. J. A. Sloane

			a(3) = 43 since 2^m contains 3 0's for m starting with 43 (2^43 = 8796093022208) and followed by 53, 61, 69, 70, 83, 87, 89, 90, 93, ...

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000079, A007377, A031147, A006889.

Cf. A063555 (analog for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063596 (for 6^k), A063606 (for 7^k), A063616 (for 8^k), A063626 (for 9^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[2^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a (* Robert G. Wilson v, Jun 12 2004 *)
nn = 100; t = Table[0, {nn}]; found = 0; k = 0; While[found < nn, k++; cnt = Count[IntegerDigits[2^k], 0]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; found++]]; t = Join[{0}, t] (* T. D. Noe, Mar 14 2012 *)

A031146(n)=for(k=0, oo, #select(d->!d, digits(2^k))==n&&return(k)) \\ M. F. Hasler, Jun 15 2018

More terms from Erich Friedman

Definition clarified by Joerg Arndt, Sep 27 2016

A063596 Least k >= 0 such that 6^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 10, 9, 13, 19, 43, 56, 41, 94, 79, 113, 100, 88, 112, 124, 127, 138, 176, 144, 175, 174, 168, 170, 210, 245, 228, 182, 237, 287, 260, 312, 321, 294, 347, 389, 365, 401, 386, 390, 419, 460, 425, 438, 426, 488, 490, 520, 458, 489, 521, 513

Offset: 0

Robert G. Wilson v, Aug 10 2001

Cf. A031146 (analog for 2^k), A063555 (for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063606 (for 7^k), A063616 (for 8^k), A063626 (for 9^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[6^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a
With[{pwr6=Table[{n,DigitCount[6^n,10,0]},{n,1000}]},Join[{0},Transpose[ Table[ SelectFirst[pwr6,#[[2]]==i&],{i,60}]][[1]]]] (* Harvey P. Dale, Dec 15 2014 *)

A063596(n)=for(k=0, oo, #select(d->!d, digits(6^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

a(0) changed to 0 (as in A031146, A063555, ...) and better title from M. F. Hasler, Jun 14 2018

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).

A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, 9, 13, 16, 9, 10, 7, 7, 9, 9, 13, 14, 15, 14, 15, 9, 9, 8, 8, 15, 11, 11, 12, 5, 12, 14, 5, 7, 14, 10, 8, 5, 16, 12

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 12 is the number of terms in A030705 and in A195945, which includes the power 7^0 = 1.

These are the row lengths of A305929. It remains an open problem to provide a proof that these rows are complete (as for 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. A030705 = row 0 of A305929: k such that 9^k has no 0's; A195945: these powers 9^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063626 = column 1 of A305929: least k such that 9^k has n digits 0 in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938 (analog for 8^k).

A305939(n,M=99*n+199,x=9)=sum(k=0,M,#select(d->!d,digits(x^k))==n)

A305939_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]++);a[^-1]}

A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, 186, 194, 191, 219, 208, 247, 267, 199, 314, 292, 263, 319, 303, 307, 345, 431, 401, 375, 388, 413, 498, 488, 504, 465, 513, 565, 464, 481, 541, 568, 532, 588, 542, 600, 677, 649, 633, 613, 734, 627

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030705: exponents of powers of 9 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. A063626: least k such that 9^k has n digits 0 in base 10.
Cf. A305939: number of k's such that 9^k has n digits 0.
Cf. A305929: row n lists exponents of 9^k with n digits 0.
Cf. A030705: { k | 9^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, ..., A306118: analog for 2^k, ..., 8^k.

A306119_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

Data corrected thanks to a remark by R. J. Mathar, by M. F. Hasler, Feb 11 2023

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A031146 Exponent of the least power of 2 having exactly n zeros in its decimal representation.

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A063596 Least k >= 0 such that 6^k has exactly n 0's in its decimal representation.

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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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A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

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A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

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