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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 03:09 EST 2022. Contains 358454 sequences. (Running on oeis4.)