A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

27, 43, 77, 61, 69, 119, 115, 158, 159, 168, 216, 232, 202, 198, 244, 270, 229, 274, 241, 273, 364, 283, 413, 298, 408, 341, 378, 431, 404, 403, 465, 483, 472, 454, 467, 508, 540, 575, 485, 576, 511, 623, 538, 515, 560, 655, 647, 661, 648, 639, 752

Offset: 0

Author

M. F. Hasler, Jun 22 2018

Keywords

• nonn
• base

Comments

a(0) is the largest term in A030704: exponents of powers of 8 without digit 0 in base 10.

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, Mar 07 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. A063616: least k such that 8^k has n digits 0 in base 10.

Cf. A305938: number of k's such that 8^k has n digits 0.

Cf. A305928: row n lists exponents of 8^k with n digits 0.

Cf. A030704: { k | 8^k has no digit 0 } : row 0 of the above.

Cf. A020665: largest k such that n^k has no digit 0 in base 10.

Cf. A071531: least k such that n^k contains a digit 0 in base 10.

Cf. A103663: least x such that x^n has no digit 0 in base 10.

Cf. A306112, ..., A306119: analog for 2^k, ..., 9^k.

Programs

PARI

A306118_vec(nMax,M=99*nMax+199,x=8,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}