A305925 -id:A305925 - cp's OEIS frontend

A008839 Numbers k such that the decimal expansion of 5^k contains no zeros.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58

Offset: 1

Probably 58 is last term.

Searched for k up to 10^10. - David Radcliffe, Dec 27 2015

			Here is 5^58, conjecturally the largest power of 5 that does not contain a 0:
34694469519536141888238489627838134765625. - _N. J. A. Sloane_, Feb 10 2023, corrected by _Patrick De Geest_, Jun 09 2024

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014.
W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero.

Cf. A000351 (5^n).
For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A305925, A052382, A027870, A102483, A103663.

[ n: n in [0..500] | not 0 in Intseq(5^n) ]; // Vincenzo Librandi Oct 19 2012

Do[ If[ Union[ RealDigits[ 5^n ][[1]]] [[1]] != 0, Print[ n ]], {n, 0, 25000}]

for(n=0,99,vecmin(digits(5^n))&& print1(n",")) \\ M. F. Hasler, Mar 07 2014

Definition corrected and initial term 0 added by M. F. Hasler, Sep 25 2011
Further edits by M. F. Hasler, Mar 08 2014
Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

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

Original entry on oeis.org

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

M. F. Hasler, Jun 14 2018

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?

			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.

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

Cf. A030701, A063575.

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

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*)

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

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

A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7, 19, 13, 15, 13, 10, 15, 12, 15, 11, 11, 15, 10, 9, 15, 17, 16, 13, 12, 12, 11, 14, 9, 14, 15, 16, 14, 13, 14, 15, 24, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 16 is the number of terms in A008839 and in A195948, which includes the power 5^0 = 1.

These are the row lengths of A305925. 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. A030701 (= row 0 of A305925): k such that 5^k has no 0's; A195948: these powers 4^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063585 (= column 1 of A305925): least k such that 5^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305945(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(5^k))==n)

A305945_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(5^k)),nMax)]++);a[^-1]}

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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A008839 Numbers k such that the decimal expansion of 5^k contains no zeros.

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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).

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A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

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

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