A292787 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^2 in base n.

3, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 6, 16, 17, 9, 19, 5, 7, 11, 23, 9, 25, 13, 27, 8, 29, 6, 31, 32, 12, 17, 15, 9, 37, 19, 13, 16, 41, 7, 43, 12, 10, 23, 47, 16, 49, 25, 18, 13, 53, 27, 11, 8, 19, 29, 59, 16, 61, 31, 28, 64, 26, 12, 67, 17, 24, 15, 71

Offset

2,1

Comments

For n > 2, a(n) <= n-1 (and the sum of digits of a(n)^2 in base n equals a(n)).

The term a(10) = 9 belongs to A058369.

For any n > 1 and k >= 0:

- let d_n(k) be the digital sum of k in base n,

- we have d_n(1) = 1 and d_n(n-1) = d_n((n-1)^2),

- for any k such that 0 <= k <= n, we have d_n(k^2) - d_n(k) = d_n((n-k)^2) - d_n(n-k),

- for any even n > 2, d_n(n/2) != d_n((n/2)^2),

- hence, for any n > 2, either a(n) < n/2 or a(n) = n-1,

- and the scatterplot of the sequence (for n > 2) has only points in the region y < x/2 and on the line y = x-1.

Apparently (see colorized scatterplot in Links section):

- for any k > 0, a(2^k + 1) = 2^k,

- if n is odd and not of the form 2^k + 1, then a(n) <= (n-1)/2.

See also A292788 for a similar sequence involving cubes instead of squares.

Links

Table of n, a(n) for n=2..72.

Rémy Sigrist, Scatterplot of the sequence for n=2..10000

Rémy Sigrist, Colorized scatterplot of the sequence for n=2..10000

Example

For n = 8:
- 1 is a power of 8,
- d_8(2) = 2 and d_8(2^2) = 4,
- d_8(3) = 3 and d_8(3^2) = 2,
- d_8(4) = 4 and d_8(4^2) = 2,
- d_8(5) = 5 and d_8(5^2) = 4,
- d_8(6) = 6 and d_8(6^2) = 8,
- d_8(7) = 7 and d_8(7^2) = 7,
- hence a(8) = 7.

Mathematica

With[{kk = 10^3}, Table[SelectFirst[Complement[Range[2, kk], n^Range@ Floor@ Log[n, kk]], Total@ IntegerDigits[#, n] == Total@ IntegerDigits[#^2, n] &] /. k_ /; MissingQ@ k -> -1, {n, 2, 72}]] (* 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^2, n), return (k))))

Crossrefs

Cf. A058369, A292788.

Sequence in context: A035634 A091563 A161985 * A335944 A035366 A114751

Adjacent sequences: A292784 A292785 A292786 * A292788 A292789 A292790

Keyword

nonn,base

Author

Rémy Sigrist, Sep 23 2017

Status

approved