A305931 -id:A305931 - cp's OEIS frontend

A305934 Powers of 3 that have exactly one digit '0' in base 10.

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 847288609443, 68630377364883, 328256967394537077627, 26588814358957503287787, 717897987691852588770249, 6461081889226673298932241, 1144561273430837494885949696427, 22528399544939174411840147874772641, 67585198634817523235520443624317923

Offset: 1

M. F. Hasler (following a suggestion by Zak Seidov), Jun 14 2018

Motivated by A030700: decimal expansion of 3^n contains no zeros (probably finite).
It appears that this sequence is finite. Is a(15) = 3^73 the last term?
There are no more terms through at least 3^(10^7) (which is a 4771213-digit number). It seems nearly certain that no power of 3 containing this many or more decimal digits could have fewer than two '0' digits. (Among numbers of the form 3^k with 73 < k <= 10^7, the only one having fewer than two '0' digits among its final 200 digits is 3^5028978.) - Jon E. Schoenfield, Jun 24 2018
The first 6 terms coincide with A305931: powers of 3 having at least one digit 0, with complement A238939 (within A000244: powers of 3) conjectured to be finite, too. Then, a(7..8) = A305931(9..10), etc.

Cf. A030700: decimal expansion of 3^n contains no zeros (probably finite), A238939: powers of 3 with no digit '0' in their decimal expansion, A000244: powers of 3.
Subsequence of A305931: powers of 3 having at least one '0'.
Cf. A305933: row n = { k | 3^k has n digits '0' }.

Select[3^Range[120], DigitCount[#, 10, 0] == 1 &] (* Michael De Vlieger, Jul 01 2018 *)

for(n=1,99, #select(t->!t,digits(3^n))==1&& print1(3^n","))

a(n) = 3^A305933(1,n).

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

A305934 Powers of 3 that have exactly one digit '0' in base 10.

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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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