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

Author

Rémy Sigrist, Sep 23 2017

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A058369, A292788.

Programs

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 *)

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))))