A298607 -id:A298607 - cp's OEIS frontend

A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, 31, 30, 38, 25, 35, 35, 36, 40, 32, 40, 43, 39, 32, 30, 30, 32, 36, 39, 23, 26, 31, 37, 27, 28, 33, 39, 28, 44, 34, 27, 43, 33, 27, 32, 31, 27, 27, 32, 35, 34, 36, 28, 32, 39, 38, 40, 28, 43, 38, 32, 22

Offset: 0

M. F. Hasler, Jun 21 2018

a(0) = 36 is the number of terms in A007377 and in A238938, which includes the power 2^0 = 1.
These are the row lengths of A305932. 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.
The average of the first 100000 terms is ~33.219 with a minimum of 12 and a maximum of 61. - Hans Havermann, Apr 26 2020

Hans Havermann, Table of n, a(n) for n = 0..10000
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.)

Row lengths of A305932 (row n = exponents of 2^k with n '0's).
Cf. A007377 = {k | 2^k has no digit 0}; A238938: powers of 2 with no digit 0.
Cf. A298607: powers of 2 with the digit '0' in their decimal expansion.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A071531: least r such that n^r has a digit 0, in base 10.
Cf. A306112: largest k such that 2^k has n digits 0, in base 10.

A305942(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(2^k))==n)

A305942_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(2^k)),nMax)]++);a[^-1]}

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]))

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A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

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

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