A336206 -id:A336206 - cp's OEIS frontend

A336207 Lexicographically earliest sequence of nonnegative terms such that whenever a(k_1) = ... = a(k_m) with k_1 < ... < k_m, the sum k_1 + ... + k_m can be computed without carries in factorial base.

0, 0, 1, 2, 3, 0, 2, 0, 4, 5, 6, 1, 5, 4, 7, 8, 9, 3, 10, 11, 12, 13, 14, 0, 8, 1, 11, 10, 15, 0, 16, 7, 17, 16, 18, 2, 19, 17, 20, 21, 22, 15, 23, 24, 25, 26, 27, 0, 13, 12, 24, 19, 28, 1, 21, 20, 29, 30, 31, 6, 30, 29, 32, 33, 34, 28, 35, 36, 37, 38, 39, 2

Offset: 1

Rémy Sigrist, Jul 12 2020

			In factorial base:
- 1 = "1", 2 = "10", 3 = "11", 4 = "20",
- we can add without carry 1 and 2, so a(1) = a(2) = 0,
- 1 + 2 + 3 implies a carry, so a(3) = 1,
- 1 + 2 + 4 and 3 + 4 imply a carry, so a(4) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation
Rémy Sigrist, C program for A336207

See A279125 and A336206 for similar sequences.

Cf. A279732.

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a(n) = 0 iff n belongs to A279732.

A383412 Lexicographically earliest sequence of integers >= 2 such that whenever a(k_1) = ... = a(k_m) with k_1 < ... < k_m, the sum k_1 + ... + k_m can be computed without carries in base a(k_1).

Original entry on oeis.org

2, 2, 2, 3, 2, 3, 4, 5, 2, 3, 5, 6, 6, 7, 7, 8, 2, 4, 9, 9, 4, 7, 9, 10, 8, 5, 5, 3, 11, 12, 5, 10, 2, 10, 11, 11, 3, 12, 12, 12, 13, 13, 6, 13, 13, 14, 14, 14, 15, 7, 15, 15, 16, 16, 16, 17, 14, 17, 18, 18, 15, 18, 19, 19, 2, 20, 20, 20, 4, 17, 17, 21, 6, 18

Offset: 0

Rémy Sigrist, Apr 26 2025

This sequence is a variant of A279125 and A336206 exploiting several bases.
This sequence is unbounded:
- by contradiction, suppose that a(n) <= B for some B >= 2,
- let U_B = {1 + B!*k, k >= 0},
- the base b expansion of any term of U_B ends with a digit 1 in any base b in the interval 2..B,
- by the pigeonhole principle, for some b in the interval 2..B, we have a(u) = b for infinitely many terms of U_B,
- however we can at most add b-1 such terms in base b, a contradiction.

			The first terms, in decimal and in base a(n), alongside the corresponding sums of indices k <= n such that a(k) = a(n) in base a(n), are:
  n   a(n)  n in base a(n)  Sums in base a(n)
  --  ----  --------------  -----------------
   0     2  0               0
   1     2  1               1
   2     2  1,0             1,1
   3     3  1,0             1,0
   4     2  1,0,0           1,1,1
   5     3  1,2             2,2
   6     4  1,2             1,2
   7     5  1,2             1,2
   8     2  1,0,0,0         1,1,1,1
   9     3  1,0,0           1,2,2
  10     5  2,0             3,2
  11     6  1,5             1,5
  12     6  2,0             3,5
  13     7  1,6             1,6
  14     7  2,0             3,6
  15     8  1,7             1,7

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A131577, A279125, A336206, A375776.

{   t = [0, 0];
    for (n = 0, 73,
        for (b = 2, oo,
            if (#t < b,
                t = concat(t, vector(#t)););
            if (sumdigits(t[b]+n, b) == sumdigits(t[b], b) + sumdigits(n, b),
                print1 (b", "); t[b] += n; break;););); }

a(n) = 2 iff n belongs to A131577.

cp's OEIS Frontend

A336207 Lexicographically earliest sequence of nonnegative terms such that whenever a(k_1) = ... = a(k_m) with k_1 < ... < k_m, the sum k_1 + ... + k_m can be computed without carries in factorial base.

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