A360964 -id:A360964 - cp's OEIS frontend

A360959 Order the nonnegative integers by increasing number of digits in base 2, then by decreasing number of digits in base 3, then by increasing number of digits in base 4, etc.

0, 1, 3, 2, 5, 7, 6, 4, 9, 11, 13, 15, 14, 12, 10, 8, 27, 29, 31, 30, 28, 25, 26, 17, 19, 21, 23, 24, 22, 20, 18, 16, 33, 35, 34, 32, 49, 51, 53, 55, 57, 59, 61, 63, 62, 60, 58, 56, 54, 52, 50, 37, 39, 41, 43, 45, 47, 48, 46, 44, 42, 40, 38, 36, 125, 127, 126

Offset: 0

Rémy Sigrist, Feb 27 2023

We ignore leading zeros.
This sequence is a permutation of the nonnegative integers with inverse A360960.
The order of appearance of two distinct integers, say x and y with x > y, depends on the parity of A360964(x, y): even implies x appears after y, odd implies x appears before y.

			The first terms, alongside their number of digits in small bases, are:
  n   a(n)  w2  w3  w4  w5  w6  w7  w8  w9  w10  w11  w12  w13  w14  w15
  --  ----  --  --  --  --  --  --  --  --  ---  ---  ---  ---  ---  ---
   0     0   0
   1     1   1
   2     3   2   2
   3     2   2   1
   4     5   3   2   2   2   1
   5     7   3   2   2   2   2   2
   6     6   3   2   2   2   2   1
   7     4   3   2   2   1
   8     9   4   3   2   2   2   2   2   2    1
   9    11   4   3   2   2   2   2   2   2    2    2    1
  10    13   4   3   2   2   2   2   2   2    2    2    2    2    1
  11    15   4   3   2   2   2   2   2   2    2    2    2    2    2    2
  12    14   4   3   2   2   2   2   2   2    2    2    2    2    2    1
  13    12   4   3   2   2   2   2   2   2    2    2    2    1
  14    10   4   3   2   2   2   2   2   2    2    1
  15     8   4   2

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

See A360982 for a similar sequence.

Cf. A360960 (inverse), A360964.

PARI
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See Links section.
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a(n) < 2^k for any n < 2^k.

A360963 Triangle T(n, k), n > 0, k = 0..n-1, read by rows: T(n, k) is the least e > 0 such that the binary expansions of n^e and k^e have different lengths.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3

Offset: 1

Rémy Sigrist, Feb 27 2023

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

			Triangle T(n, k) begins:
  n\k | 0  1  2  3  4  5  6  7  8  9  10  11  12  13  14
  ----+-------------------------------------------------
    1 | 1
    2 | 1  1
    3 | 1  1  2
    4 | 1  1  1  1
    5 | 1  1  1  1  4
    6 | 1  1  1  1  2  2
    7 | 1  1  1  1  2  2  3
    8 | 1  1  1  1  1  1  1  1
    9 | 1  1  1  1  1  1  1  1  6
   10 | 1  1  1  1  1  1  1  1  4  4
   11 | 1  1  1  1  1  1  1  1  3  3   3
   12 | 1  1  1  1  1  1  1  1  2  2   2   2
   13 | 1  1  1  1  1  1  1  1  2  2   2   2   3
   14 | 1  1  1  1  1  1  1  1  2  2   2   2   3   4
   15 | 1  1  1  1  1  1  1  1  2  2   2   2   3   4   6

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)
Rémy Sigrist, Colored representation of the first 512 rows

Cf. A029837, A183200, A360964.

T(n,k) = { for (e=1, oo, if (#binary(n^e) != #binary(k^e), return (e))) }

T(n, 0) = 1.

T(n, n-1) = A183200(n-1) for n > 1.

cp's OEIS Frontend

A360959 Order the nonnegative integers by increasing number of digits in base 2, then by decreasing number of digits in base 3, then by increasing number of digits in base 4, etc.

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