A375969 Lexicographically earliest sequence of nonnegative integers with distinct digit averages.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 29, 49, 69, 89, 100, 101, 103, 104, 106, 107, 109, 119, 139, 149, 169, 179, 199, 299, 499, 599, 799, 899, 1000, 1002, 1004, 1006, 1008, 1019, 1039, 1059, 1079, 1099, 1299, 1499, 1699, 1899, 2999, 4999, 6999

Offset: 1

Author

Rémy Sigrist, Sep 04 2024

Keywords

• nonn
• base

Comments

Equivalently, fixed points of A375968.

The sequence {A007953(a(n)) / A055642(a(n)), n > 0} runs uniquely through every rational number between 0 and 9.

Examples

			The first terms, alongside the corresponding digit average, are:
  n   a(n)  Digit average
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     3              3
   5     4              4
   6     5              5
   7     6              6
   8     7              7
   9     8              8
  10     9              9
  11    10            1/2
  12    12            3/2
  13    14            5/2
  14    16            7/2
  15    18            9/2
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10459
• Rémy Sigrist, PARI program

Crossrefs

Cf. A007953, A051885, A055642, A375968.

Programs

Mathematica

kmax=7000; avg={}; list={}; For[k=0, k<=kmax, k++,mn=Mean[IntegerDigits[k]]; If[!MemberQ[avg,mn], AppendTo[avg, mn]; AppendTo[list, k]]]; list (* Stefano Spezia, Sep 07 2024 *)

PARI

avg(n, base = 10) = { my (d = digits(n, base)); vecsum(d) / max(1, #d) }
{ V = Map(); k = 0; for (n = 0, 6999, v = avg(n); if (!mapisdefined(V, v), mapput(V, v, n); print1 (n", "););); }

PARI

\\ See Links section.

Python

from math import gcd
from itertools import count, islice
def agen():
    m = 0
    yield 0
    for w in count(1): # w = number of digits
        for s in range(1, 9*w+1): # s = sum of digits
            if gcd(s, w) == 1:
                d, r = [1] + [0 for _ in range(w-1)], s-1
                for k in range(w-1, -1, -1):
                    d[k], r = d[k] + min(r, 9), r - min(r, 9)
                yield int("".join(map(str, d)))
print(list(islice(agen(), 54)))
# Michael S. Branicky, Sep 08 2024 after Rémy Sigrist PARI in link