A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7, 19, 13, 15, 13, 10, 15, 12, 15, 11, 11, 15, 10, 9, 15, 17, 16, 13, 12, 12, 11, 14, 9, 14, 15, 16, 14, 13, 14, 15, 24, 14

Offset

0,1

Comments

a(0) = 16 is the number of terms in A008839 and in A195948, which includes the power 5^0 = 1.

These are the row lengths of A305925. It remains an open problem to provide a proof that these rows are complete (as are 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) A305945(n, M=99*n+199)=sum(k=0, M, #select(d->!d, digits(5^k))==n)
(PARI) A305945_vec(nMax, M=99*nMax+199, a=vector(nMax+=2))={for(k=0, M, a[min(1+#select(d->!d, digits(5^k)), nMax)]++); a[^-1]}

Crossrefs

Cf. A030701 (= row 0 of A305925): k such that 5^k has no 0's; A195948: these powers 4^k.

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

Cf. A063585 (= column 1 of A305925): least k such that 5^k has n digits '0' in base 10.

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

Sequence in context: A173778 A103908 A192632 * A010855 A226757 A248764

Adjacent sequences: A305942 A305943 A305944 * A305946 A305947 A305948

Keyword

nonn,base

Author

M. F. Hasler, Jun 22 2018

Status

approved