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

Author

Michael De Vlieger, Jan 02 2025

Keywords

• nonn
• tabf
• easy

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.

Examples

			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.
		

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

Crossrefs

Cf. A019565.

Programs

Mathematica

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

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.