A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, 1091, 1239, 1201, 1224, 1210, 1141, 1339, 1240, 1282, 1395, 1449, 1416, 1408, 1616, 1524, 1727, 1725, 1553, 1942, 1907, 1945, 1870, 1724, 1972, 1965, 2075, 1983, 2114, 2257, 2256

Offset: 0

Author

M. F. Hasler, Jun 22 2018

Keywords

• nonn
• base

Comments

a(0) is the largest term in A007377: exponents of powers of 2 without digit 0.

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.

Links

• 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

Cf. A031146: least k such that 2^k has n digits 0 in base 10.

Cf. A305942: number of k's such that 2^k has n digits 0.

Cf. A305932: row n lists exponents of 2^k with n digits 0.

Cf. A007377: { k | 2^k has no digit 0 } : row 0 of the above.

Cf. A238938: { 2^k having no digit 0 }.

Cf. A027870: number of 0's in 2^n (and A065712, A065710, A065714, A065715, A065716, A065717, A065718, A065719, A065744 for digits 1 .. 9).

Cf. A102483: 2^n contains no 0 in base 3.

Programs

PARI

A306112_vec(nMax,M=99*nMax+199,x=2,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}