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A286512 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. 0
17, 31, 63, 86, 91, 103, 118, 133, 155, 157, 211, 270, 290, 301, 338, 352, 421, 432, 440, 441, 450, 478, 513, 533, 693, 853, 1051, 1237, 1363, 1459, 1526, 1665, 2781 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..33.


(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.


Cf. A247889, A007953.

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.)

Sequence in context: A267781 A270441 A256374 * A087166 A167496 A164041

Adjacent sequences:  A286509 A286510 A286511 * A286513 A286514 A286515




M. F. Hasler, May 18 2017



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Last modified July 25 19:05 EDT 2021. Contains 346291 sequences. (Running on oeis4.)