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

Author

M. F. Hasler, Jun 21 2018

Keywords

• nonn
• base

Comments

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

Links

• 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.)

Crossrefs

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.

Programs

PARI

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

PARI

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]}