cp's OEIS Frontend

Proposed by Mark Sapir, Math. Dept., Vanderbilt University, who remarks (August 2000) that he can prove that a(n) is always finite and that a(1) = 2.

Values of a(n) for n>1 computed numerically by Michael Kleber, Sep 02 2000 and David W. Wilson, Sep 06 2000.

All terms except the first are only conjectures. For the theorem that a(n) is always finite, see Senge-Straus and Stewart. - N. J. A. Sloane, Jan 06 2011

The set of these numbers appears to be finite, and probably 2781 is its largest element.

The motivation for this sequence is the study of the behavior of the sum of digits of powers of a given number. Statistically, sumdigits(n^k) ~ 4.5*log_10(n')*k (where n' = n without trailing 0's), but typically fluctuations of some percent persist up to large values of k. (Cf. the graph of sequences n^k cited in the cross-references.)

The ratio of 11/10 is somewhat arbitrary, but larger ratios of the simple form (1 + 1/m) yield quite small subsets of this sequence (for m=2 the only element is 118, for m=3 the set is {31, 86, 118}, for m=1 it is empty), and smaller ratios yield much larger (possibly infinite?) sets. Also, the condition can be written sumdigits(N^k)-N > N/10, and 10 is the base we are using.

To compute the sequence A247889 we would like to have a rule telling us when we can stop the search for an exponent. It appears that sumdigits(N^k) >= 2*N is a limit that works for all N; the present sequence gives counterexamples to the (r.h.s.) limit of 1.1*N. The above comment mentioned the counterexamples {118} resp. {31, 86, 118}) for limits N*3/2 and N*4/3.

118 is exceptional in the sense that it appears to be the only number m for which the smallest k such that sumdigits(m^k) = m occurs after the smallest k such that sumdigits(m^k) > m*3/2. If this last limit is decreased to m*4/3, then 31 and 86 also have this property. It appears that no number has this property if the limit is increased to 2m, see also A247889.

It is also remarkable that many values in the sequence are repeated (19, 55, 64, 109, 190, 154, 280, 289, 334 (3 times), 379, 406, 424, ...), while most other numbers never appear.

Corresponding r's: any, 4, 3, 2, 6, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3.