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.

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

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 + (n-k-e) 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

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Rémy Sigrist, Colorized scatterplot of the sequence for n=2..100000

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

Cf. A053735, A070276, A292787.

Sequence in context: A202568 A145685 A333951 * A138597 A206758 A019290

Adjacent sequences: A292785 A292786 A292787 * A292789 A292790 A292791

Keyword

nonn,base,look

Author

Rémy Sigrist, Sep 23 2017

Status

approved