A305943 - OEIS

A305943

Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 16, 17, 15, 30, 24, 32, 24, 12, 16, 16, 23, 23, 20, 23, 19, 23, 10, 21, 20, 21, 23, 20, 19, 23, 23, 22, 16, 18, 20, 20, 13, 15, 25, 24, 28, 24, 21, 16, 14, 23, 21, 19, 23, 19, 27, 26, 22, 18, 27, 16, 31, 21, 18, 25, 24

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OFFSET

0,1

COMMENTS

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

These are the row lengths of A305933. 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 vanishingly small, cf. Khovanova link.

LINKS

Table of n, a(n) for n=0..100.

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) A305943(n, M=99*n+199)=sum(k=0, M, #select(d->!d, digits(3^k))==n)

(PARI) A305943_vec(nMax, M=99*nMax+199, a=vector(nMax+=2))={for(k=0, M, a[min(1+#select(d->!d, digits(3^k)), nMax)]++); a[^-1]}

CROSSREFS

Cf. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.

Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.

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

Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.

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

Sequence in context: A104958 A124601 A040508 * A318090 A070716 A217894

Adjacent sequences: A305940 A305941 A305942 * A305944 A305945 A305946

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jun 22 2018

STATUS

approved