

A292788


For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^3 in base n.


3



56953, 13, 2, 3, 20, 2, 6, 3, 8, 5, 1110, 3, 65, 8, 4, 7, 86, 9, 2374, 4, 8, 12, 114, 3, 99, 12, 135, 15, 3567, 4, 185, 15, 11, 16, 6, 19, 73, 20, 12, 5, 81, 6, 85, 23, 19, 24, 93, 7, 97, 24, 18, 27, 796, 28, 44, 7, 19, 28, 413, 4, 365, 32, 8, 31, 26, 21, 200
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,1


COMMENTS

The term a(10) = 8 belongs to A070276.
For any n > 1, a(n^2) <= n.
Is this sequence defined for any n > 1 ?
Apparently, a(k) < k for any odd k > 3.
Among the first 99 999 terms, the digital sum of a(n) in base n is > n for n = 2, 12, 20, 30.
The scatterplot of the sequence shows beams on the upper part, which correspond to clusters of close points for which a(n) = k*n + (nke) for some k > 0 and e in { 0, 2 }.
See also A292787 for a similar sequence involving squares instead of cubes.
The least positive k, not a power of 2, such that the hamming weight of k equals the hamming weight of k^4 is 34225258495.


LINKS



EXAMPLE

For n = 3:
 let d_3 denote the digital sum in base 3 (d_3 = A053735),
 1 is a power of 3,
 d_3(2) = 2 and d_3(2^3) = 4,
 3 is a power of 3,
 d_3(4) = 2 and d_3(4^3) = 4,
 d_3(5) = 3 and d_3(5^3) = 7,
 d_3(6) = 2 and d_3(6^3) = 4,
 d_3(7) = 3 and d_3(7^3) = 5,
 d_3(8) = 4 and d_3(8^3) = 8,
 9 is a power of 3,
 d_3(10) = 2 and d_3(10^3) = 4,
 d_3(11) = 3 and d_3(11^3) = 9,
 d_3(12) = 2 and d_3(12^3) = 4,
 d_3(13) = 3 and d_3(13^3) = 3,
 hence a(3) = 13.


MATHEMATICA

With[{kk = 10^5}, Table[SelectFirst[Complement[Range[2, kk], n^Range@ Floor@ Log[n, kk]], Total@ IntegerDigits[#, n] == Total@ IntegerDigits[#^3, n] &] /. k_ /; MissingQ@ k > 1, {n, 2, 68}]] (* Michael De Vlieger, Sep 24 2017 *)


PROG

(PARI) a(n) = my (p=1); for (k=1, oo, if (k==p, p*=n, if (sumdigits(k, n) == sumdigits(k^3, n), return (k))))


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



