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

0,1

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

Table of n, a(n) for n=0..50.

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.)

Prog

(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]}

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.

Sequence in context: A097693 A184087 A304518 * A184086 A113690 A043379

Adjacent sequences: A306109 A306110 A306111 * A306113 A306114 A306115

Keyword

nonn,base

Author

M. F. Hasler, Jun 22 2018

Status

approved