A360963 -id:A360963 - cp's OEIS frontend

A360964 Triangle T(n, k), n > 0, k = 0..n-1, read by rows: T(n, k) is the least base b >= 2 where the number of digits of n and k are different.

2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 6, 2, 2, 2, 2, 5, 6, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 10, 2, 2, 2, 2, 2, 2, 2, 2, 3, 10, 11, 2, 2, 2, 2, 2, 2, 2, 2, 3, 10, 11, 12, 2, 2, 2, 2, 2, 2, 2, 2, 3, 10, 11, 12, 13

Offset: 1

Rémy Sigrist, Feb 27 2023

Leading zeros are ignored (and 0 is assumed to have length 0).

			Triangle T(n, k) begins:
  n\k | 0  1  2  3  4  5  6  7  8   9  10  11
  ----+--------------------------------------
    1 | 2
    2 | 2  2
    3 | 2  2  3
    4 | 2  2  2  2
    5 | 2  2  2  2  5
    6 | 2  2  2  2  5  6
    7 | 2  2  2  2  5  6  7
    8 | 2  2  2  2  2  2  2  2
    9 | 2  2  2  2  2  2  2  2  3
   10 | 2  2  2  2  2  2  2  2  3  10
   11 | 2  2  2  2  2  2  2  2  3  10  11
   12 | 2  2  2  2  2  2  2  2  3  10  11  12

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, Colored representation of the first 512 rows

Cf. A052410, A360963.

T(n,k) = { for (b=2, oo, if (#digits(n,b) != #digits(k,b), return (b))) }

T(n, 0) = 2.

T(n, n-1) = A052410(n) for any n > 1.

A360982 Order the nonnegative integers by increasing binary length of values, then by decreasing binary length of values squared, then by increasing binary length of values cubed, etc.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 15, 14, 13, 10, 9, 8, 11, 23, 24, 25, 27, 30, 31, 29, 28, 26, 20, 18, 17, 16, 19, 21, 22, 46, 47, 48, 50, 49, 54, 55, 59, 60, 61, 63, 62, 58, 56, 57, 52, 51, 53, 39, 40, 36, 35, 34, 32, 33, 37, 38, 42, 41, 43, 44, 45, 91, 92, 93, 94

Offset: 0

Rémy Sigrist, Feb 27 2023

We ignore leading zeros (and 0 is assumed to have binary length 0).
This sequence is a permutation of the nonnegative integers with inverse A360983.
The order of appearance of two distinct integers, say x and y with x > y, depends on the parity of A360963(x, y): even implies x appears before y, odd implies x appears after y.

			The first terms, alongside the binary length of their first powers, are:
  n   a(n)  w1  w2  w3  w4  w5  w6
  --  ----  --  --  --  --  --  --
   0     0   0
   1     1   1
   2     3   2   4
   3     2   2   3
   4     6   3   6   8
   5     7   3   6   9
   6     5   3   5   7  10
   7     4   3   5   7   9
   8    12   4   8  11
   9    15   4   8  12  16  20  24
  10    14   4   8  12  16  20  23
  11    13   4   8  12  15
  12    10   4   7  10  14
  13     9   4   7  10  13  16  20
  14     8   4   7  10  13  16  19
  15    11   4   7  11

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of the first 2^15 terms
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A360959 for a similar sequence.

Cf. A360963, A360983 (inverse).

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A360964 Triangle T(n, k), n > 0, k = 0..n-1, read by rows: T(n, k) is the least base b >= 2 where the number of digits of n and k are different.

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