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Numbers N for which there is k > 0 such that sum of digits(N^k) = N, but the least such k is larger than the least k for which sum of digits(N^k) > N*11/10.
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%I #28 Feb 27 2020 03:03:12

%S 17,31,63,86,91,103,118,133,155,157,211,270,290,301,338,352,421,432,

%T 440,441,450,478,513,533,693,853,1051,1237,1363,1459,1526,1665,2781

%N Numbers N for which there is k > 0 such that sum of digits(N^k) = N, but the least such k is larger than the least k for which sum of digits(N^k) > N*11/10.

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

%C 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.)

%C 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.

%C 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.

%o (PARI) for(n=1,5000,A247889(n)&&!A247889(n,n*11\10)&&print1(n",")) \\ Here, A247889() is a variant of the function computing that sequence which accepts as second optional argument a limit m, stopping the search for the exponent as soon as the digital sum of n^k exceeds m.

%Y Cf. A247889, A007953.

%Y Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12). (In these sequences, k is fixed and n is the index/exponent; in the present sequence it's the opposite and therefore the names k <-> n are exchanged.)

%K nonn,base,fini,full

%O 1,1

%A _M. F. Hasler_, May 18 2017