A066005 -id:A066005 - cp's OEIS frontend

A358633 a(n) is the smallest k > 1 such that the sum of digits of n^k is a power of n (or -1 if no such k exists).

2, 2, 2, 18, 8, 7, 4, 3, 2, 2, 45741764, 4216, 32, 537, 39, 44, 3, 3, 1187, 13, 67, 4

Offset: 1

Jon E. Schoenfield, Nov 24 2022

a(11), if it is not -1, seems likely to exceed 4*10^6.
Additional known terms: a(12)..a(22) = {4216, 32, 537, 39, 44, 3, 3, 1187, 13, 67, 4}; a(24)..a(28) = {88, 4, 3, 3, 4}; a(30) = 399, a(31) = 7, a(33)..a(55) = {159, 7, 5, 4, 191, 188, 228, 13, 389332, 236, 7, 11543, 6, 5, 302, 292, 15405, 788, 337, 18213, 7, 6, 21248}; a(57)..a(60) = {413, 7, 25683, 1044}; a(62) = 476.
a(10^m) = 2 for all m >= 0, since the sum of digits of (10^m)^2 is 1.
If n is not divisible by 10, then a(n) tends to be fairly close to a number x such that the number of digits of n^x is (2/9)*n^j for some positive integer j, i.e., log_10(n^x) ~ (2/9)*n^j, so a(n) ~ (2/9)*n^j/log_10(n) for some integer j. E.g., a(12) = 4216 ~ 4269.90... = (2/9)*12^4/log_10(12). For n = 11, such numbers x are (2/9)*11^j/log_10(11) = 0.213389... * 11^j, which, for j = 1..7, round to 2, 26, 284, 3124, 34366, 378032, and 4158357. (By exhaustive search, a(n) > 10^5 (or a(n) = -1) for n = 23, 29, 32, 56, and 61, and (if a(11) != -1) a(11) > 4*10^5, so a(11) seems very likely to be either in the general vicinity of 4.16*10^6 or > 4.5*10^7.)
a(32) = 4950773; a(61) = 1722427. - Martin Ehrenstein, Nov 25 2022
3*10^7 < a(23) <= 555650815. - Martin Ehrenstein, Nov 28 2022

			The sum of digits of 1^2 = 1 is 1, which is a power of 1, so a(1) = 2.
The sum of digits of 2^2 = 4 is 4, which is a power of 2, so a(2) = 2.
For k = 2..17, the sum of digits of 4^k is {7, 10, 13, 7, 19, 22, 25, 19, 31, 25, 37, 40, 43, 37, 58, 61}, none of which is a power of 4, but the sum of digits of 4^18 = 68719476736 is 6+8+7+1+9+4+7+6+7+3+6 = 64 = 4^3, so a(4) = 18.

Cf. A095412, A118872.

Cf. A066005 (sum of digits of 11^n).

isok(k,n) = my(s=sumdigits(n^k), x, y); if (s==1, return(2)); (s==n) || ((ispower(s,,&x)) && ((x==n) || (ispower(n,,&y) && (y==x))));
a(n) = my(k=2); while (!isok(k,n), k++); k; \\ Michel Marcus, Nov 25 2022

a(11)-a(22) from Martin Ehrenstein, Nov 26 2022

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.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, May 18 2017

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.

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.)

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.

A287058 Sum of decimal digits of 118^n.

Original entry on oeis.org

1, 10, 19, 19, 55, 64, 55, 64, 82, 91, 109, 100, 109, 181, 118, 145, 127, 163, 154, 172, 154, 190, 226, 190, 208, 217, 271, 289, 253, 280, 298, 307, 334, 289, 334, 280, 361, 343, 334, 379, 406, 406, 379, 424, 379, 424, 415, 406, 523, 433, 478

Offset: 0

M. F. Hasler, May 18 2017

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.

Cf. A007953, A247889.

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).

Total[IntegerDigits[#]]&/@NestList[118#&,1,50] (* Harvey P. Dale, Feb 24 2022 *)

PARI
```
a(n)=sumdigits(118^n)
```

cp's OEIS Frontend

A358633 a(n) is the smallest k > 1 such that the sum of digits of n^k is a power of n (or -1 if no such k exists).

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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.

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A287058 Sum of decimal digits of 118^n.

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