A305934 -id:A305934 - cp's OEIS frontend

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68, 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73, 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136, 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129

Offset: 0

M. F. Hasler, Jun 14 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 (3, 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, just 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 longer guaranteed to get a partition of the integers.
The author finds the idea of partitioning the integers in this elementary yet highly nontrivial way appealing, as is the fact that the initial rows are just roughly one line long. Will this property continue to hold for large n, or if not, how will the row lengths evolve?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68 (cf. A030700)
1 : 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73
2 : 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136
3 : 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129
4 : 35, 59, 65, 66, 70, 74, 77, 79, 88, 98, 106, 116, 117, 128, 130, 131, 197, 205
5 : 57, 76, 78, 80, 86, 89, 91, 92, 101, 102, 104, 109, 115, 118, 122, 127, 134, 135, 164, 166, 203, 212, 237
...
The first column is A063555: least k such that 3^k has n digits '0' in base 10.
Row lengths are 23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, ... (A305943).
Last term of the rows (i.e., largest k such that 3^k has exactly n digits 0) are (68, 73, 136, 129, 205, 237, 317, 268, 251, 276, 343, ...), A306113.
Inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 10, 11, 12, 13, 24, 25, 26, 27, 14, 28, 69, 38, 15, 16, 29, 17, 18, 19, 30, 39, 20, ...), not in OEIS.

Cf. A030700, A063555.
Cf. A305932 (analog for 2^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
Cf. A305934: powers of 3 with exactly one '0', A305943: powers of 3 with at least one '0'.

apply( A305933_row(n,M=50*n+70)=select(k->#select(d->!d,digits(3^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 elements, and inverse permutation.

A305931 Powers of 3 having at least one digit '0' in their decimal representation.

Original entry on oeis.org

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 10460353203, 31381059609, 847288609443, 68630377364883, 205891132094649, 1853020188851841, 5559060566555523, 50031545098999707, 150094635296999121, 450283905890997363, 1350851717672992089, 4052555153018976267, 12157665459056928801

Offset: 1

M. F. Hasler, Jun 15 2018

The analog of A298607 for 3^k instead of 2^k.

The complement A238939 is conjectured to have only 23 elements, the largest being 3^68. Thus, all larger powers of 3 are (conjectured to be) in this sequence. Each of the subsequences "powers of 3 with exactly n digits 0" is conjectured to be finite. Provided there is at least one such element for each n >= 0, this leads to a partition of the integers, given in A305933.

Cf. A030700 = row 0 of A305933: decimal expansion of 3^n contains no zeros.
Complement (within A000244: powers of 3) of A238939: powers of 3 with no digit '0' in their decimal expansion.
Analog of A298607: powers of 2 with the digit '0' in their decimal expansion.
The first six terms coincide with the finite sequence A305934: powers of 3 having exactly one digit 0.

Select[3^Range[0,40],DigitCount[#,10,0]>0&] (* Harvey P. Dale, May 30 2020 *)

for(k=0,69, vecmin(digits(3^k))|| print1(3^k","))

select( t->!vecmin(digits(t)), apply( k->3^k, [0..40]))

A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 16, 17, 15, 30, 24, 32, 24, 12, 16, 16, 23, 23, 20, 23, 19, 23, 10, 21, 20, 21, 23, 20, 19, 23, 23, 22, 16, 18, 20, 20, 13, 15, 25, 24, 28, 24, 21, 16, 14, 23, 21, 19, 23, 19, 27, 26, 22, 18, 27, 16, 31, 21, 18, 25, 24

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

These are the row lengths of A305933. 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 vanishingly small, 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. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.
Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305943(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(3^k))==n)

A305943_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(3^k)),nMax)]++);a[^-1]}

A305930 Number of digits '0' in 3^n (in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 0, 1, 0, 0, 0, 1, 2, 0, 2, 2, 0, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 1, 2, 7, 6, 2, 5, 2, 4, 2, 2, 2, 1, 2, 4, 4, 3, 0, 2, 4, 2, 1, 1, 4, 3, 5, 4, 5, 4, 5, 3, 3, 2, 6, 6, 5, 3, 4, 5, 3, 5, 5, 2, 6, 6, 2, 6, 4, 7

Offset: 0

M. F. Hasler, Jun 22 2018

			3^10 = 59049 is the smallest power of 3 having a digit 0, so a(10) = 1 is the first nonzero term.

Cf. A027870 (analog for 2^k), A030700 (indices of zeros).
Cf. A063555: index of first appearence of n in this sequence.
Cf. A305933: table with n in row a(n).

Haskell
```
a305930 = a055641 . a000244
```

Table[ Count[ IntegerDigits[3^n], 0], {n, 0, 100} ]
DigitCount[3^Range[0,110],10,0]

apply( A305930(n)=#select(d->!d,digits(3^n)), [0..99])

a(n) = A055641(A000244(n)).

a(A030700(n)) = 0; a(A305934(n)) = 1; a(A305931(n)) >= 1; a(A305933(n,k)) = n.

cp's OEIS Frontend

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

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A305931 Powers of 3 having at least one digit '0' in their decimal representation.

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A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

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A305930 Number of digits '0' in 3^n (in base 10).

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