A287076 a(n) = least k > n with the same sum of digits as n in some base b > 1.

2, 4, 5, 6, 6, 9, 10, 12, 10, 12, 13, 16, 14, 16, 19, 20, 18, 20, 21, 22, 22, 24, 25, 30, 26, 28, 29, 30, 30, 36, 33, 34, 34, 36, 37, 40, 38, 40, 42, 42, 42, 44, 45, 48, 46, 48, 49, 56, 50, 52, 53, 56, 54, 57, 57, 58, 58, 60, 61, 64, 62, 66, 67, 66, 66, 68, 69

Offset: 1

Author

Rémy Sigrist, May 19 2017

Keywords

• nonn
• base

Comments

More formally: a(n) = Min_{b>1} f_b(n), where f_b(n) = least k > n with the same sum of digits as n in base b.

We have the following properties:

- f_b(b) = b^2 for any b > 1,

- f_b(b^k) = b^(k+1) for any b > 1 and k >= 0,

- f_b(n) = b + n - 1 for any b > 1 and n < b,

- f_b(n) - n >= b - 1 for any b > 1 and n > 0.

Also, f_2 = A057168 and f_10 = A228915.

For any n > 0, n < a(n) <= 2*n.

Conjecturally, a(n) ~ n.

The derived sequence e(n) = a(n) - n is unbounded: for any n > 0:

- for any b such that 1 < b <= n, let x_b = the least power of b such that f_b(i*x_b) - i*x_b >= n for any i > 0,

- let X = Lcm_{b=2..n} x_b,

- then f_b(X) - X >= n for any b such that 1 < b <= n,

- also, f_b(X) - X >= b - 1 >= n for any b > n,

- hence a(X) - X = e(X) >= n, QED.

Examples

			The following table shows f_b(8) for all bases b > 1:
b    f_b(8)   8 in base b   f_b(8) in base b
--   ------   -----------   ----------------
2        16        "1000"            "10000"
3        14          "22"              "112"
4        17          "20"              "101"
5        12          "13"               "22"
6        13          "12"               "21"
7        14          "11"               "20"
8        64          "10"              "100"
b>8     b+7           "8"               "17"
Hence, a(8) = f_5(8) = 12.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A287076

Crossrefs

Cf. A057168, A228915.