A305946 Number of powers of 6 having exactly n digits '0' (in base 10), conjectured.

14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, 7, 15, 14, 16, 13, 22, 12, 17, 15, 17, 7, 6, 14, 22, 13, 19, 14, 12, 15, 7, 11, 14, 6, 12, 9, 12, 9, 14, 13, 15, 21

Offset

0,1

Comments

a(0) = 14 is the number of terms in A030702 and in A195948, which includes the power 6^0 = 1.

These are the row lengths of A305926. 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.

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) A305946(n, M=99*n+199)=sum(k=0, M, #select(d->!d, digits(6^k))==n)
(PARI) A305946_vec(nMax, M=99*nMax+199, a=vector(nMax+=2))={for(k=0, M, a[min(1+#select(d->!d, digits(6^k)), nMax)]++); a[^-1]}

Crossrefs

Cf. A030702 = row 0 of A305926: k such that 6^k has no 0's; A238936: these powers 6^k.

Cf. A020665: largest k such that n^k has no '0's.

Cf. A063596 = column 1 of A305926: least k such that 6^k has n digits '0' in base 10.

Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

Sequence in context: A128486 A147370 A140739 * A349806 A259531 A240815

Adjacent sequences: A305943 A305944 A305945 * A305947 A305948 A305949

Keyword

nonn,base

Author

M. F. Hasler, Jun 22 2018

Status

approved