A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, 9, 13, 16, 9, 10, 7, 7, 9, 9, 13, 14, 15, 14, 15, 9, 9, 8, 8, 15, 11, 11, 12, 5, 12, 14, 5, 7, 14, 10, 8, 5, 16, 12

Offset

0,1

Comments

a(0) = 12 is the number of terms in A030705 and in A195945, which includes the power 7^0 = 1.

These are the row lengths of A305929. 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) A305939(n, M=99*n+199, x=9)=sum(k=0, M, #select(d->!d, digits(x^k))==n)
(PARI) A305939_vec(nMax, M=99*nMax+199, x=9, a=vector(nMax+=2))={for(k=0, M, a[min(1+#select(d->!d, digits(x^k)), nMax)]++); a[^-1]}

Crossrefs

Cf. A030705 = row 0 of A305929: k such that 9^k has no 0's; A195945: these powers 9^k.

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

Cf. A063626 = column 1 of A305929: least k such that 9^k has n digits 0 in base 10.

Cf. A305942 (analog for 2^k), ..., A305947, A305938 (analog for 8^k).

Sequence in context: A363151 A359632 A364135 * A206423 A319406 A038333

Adjacent sequences: A305936 A305937 A305938 * A305940 A305941 A305942

Keyword

nonn,base

Author

M. F. Hasler, Jun 22 2018

Status

approved