A379770 Irregular triangle T(j,k) read by rows: split the natural numbers m > 1 into groups of length 2^s and then sort each group in the order defined in A019565.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 19, 24, 21, 25, 22, 26, 23, 28, 27, 29, 30, 31, 32, 33, 34, 36, 35, 40, 37, 48, 41, 38, 42, 49, 39, 50, 44, 43, 52, 51, 45, 56, 46, 53, 57, 54, 47, 58, 55, 60, 59, 61, 62, 63, 64, 65, 66, 68, 67

Offset

0,2

Comments

In other words, function f(m) = Product_{i=0..Floor(log_2(m))} prime(i), with i the index of 1's in the binary expansion of m.

A permutation of the natural numbers.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows j = 0..13, flattened)

Michael De Vlieger, Scatterplot of a(n), n = 0..131071.

Michael De Vlieger, Log log scatterplot of a(n), n = 0..131071

Michael De Vlieger, Fan style binary tree of T(j,k), j = 0..10, k = 1..2^(j-1), i.e., a(n) = 0..2047, with a color function where k = 1 is green, k = 2^(j-1) in red, indicating displacement of a(n) from its order in the natural numbers.

Index entries for sequences that are permutations of the natural numbers

Formula

Length of row j is 2^(j-1) = A000079(j-1).

T(0,1) = a(0) = 1.

T(j,1) = a(2^(j-1)) = 2^(j-1) + 1.

T(j, 2^(j-1)) = a(2^j - 1) = 2^j.

Example

Table below shows rows j = 0..5:
j\k   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16
-------------------------------------------------------------------
0:    1;
1:    2;
2:    3,  4;
3:    5,  6,  7,  8;
4:    9, 10, 12, 11, 13, 14, 15, 16;
5:   17, 18, 20, 19, 24, 21, 25, 22, 26, 23, 28, 27, 29, 30, 31, 32;
.
These correspond with values f(T(j,k)) as shown below:
j\k   1   2   3   4   5   6    7    8    9   10   11   12   13   14    15    16
-------------------------------------------------------------------------------
0:    1;
1:    2;
2:    3,  6;
3:    5, 10, 15, 30;
4:    7, 14, 21, 35, 42, 70, 105, 210;
5:   11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310;
.
T(4,4) = a(11) = 35, while A019565(11) = 42, since 11_2 = "1011", f(11) = 2*3*7 = 42, but A019565(12) = 35 since 12_2 = "1100", f(12) = 5*7 is smaller than 42, therefore a(11) = 35, and a(12) = 42.

Mathematica

Flatten@ Table[
  SortBy[Range[2^n, 2^(n + 1) - 1],
    Times @@ Flatten@
      MapIndexed[Prime[#2]^#1 &,
        Reverse@ IntegerDigits[#, 2]] &], {n, 0, 8}]

Crossrefs

Cf. A019565.

Sequence in context: A031297 A239086 A306866 * A187769 A357492 A209862

Adjacent sequences: A379767 A379768 A379769 * A379771 A379772 A379773

Keyword

nonn,tabf,easy

Author

Michael De Vlieger, Jan 02 2025

Status

approved