

A287076


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


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
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OFFSET

1,1


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.


LINKS

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


EXAMPLE

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.


CROSSREFS

Cf. A057168, A228915.
Sequence in context: A083788 A058317 A278761 * A100006 A196262 A238461
Adjacent sequences: A287073 A287074 A287075 * A287077 A287078 A287079


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, May 19 2017


STATUS

approved



