A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

1, 2, 4, 6, 8, 12, 20, 16, 18, 40, 28, 32, 24, 80, 56, 30, 64, 36, 100, 112, 60, 42, 128, 48, 160, 196, 90, 84, 66, 256, 54, 200, 224, 120, 126, 132, 78, 512, 72, 320, 392, 150, 168, 198, 156, 88, 1024, 96, 400, 448, 180, 252, 264, 234, 176, 104, 2048, 108, 500, 784, 240, 294, 396, 312, 352, 208, 140, 4096, 144, 640, 896, 270, 336, 528, 468, 704, 416, 280, 204, 8192

Offset: 1

Author

Frank M Jackson, Dec 30 2024

Keywords

• nonn
• tabf

Comments

A permutation of the practical numbers.

This sequence is presented as an array of rows. The first row contains a single term of value 1. Subsequent rows are infinite sequences and are presented as a square array by listing the antidiagonals downwards that is 1: 2; 4,6; 8,12,20; etc.

The first column contains the primitive practical numbers A267124; each row lists all practical numbers (A005153) having the same primitive practicle progenitor and which is the first term in each row. See A379325 comments for further details. If T[1,m] is squarefree then the row is identical to the same squarefree row in A284457.

Every primitive practical number A267124(n) is the progenitor of a disjoint subsequence of the practical numbers. If the PP column represents the sequence of primitive practical numbers A267124, the table below give the 7 initial terms of the disjoint sequences of practical numbers A005153 generated by the initial 7 terms of the sequence of primitive practical numbers.

PP: Disjoint subsequence of A005153

-- -------------------------------

1: 1

2: 2, 4, 8, 16, 32, 64,128, . . .- A000079 with offset 1,1

6: 6, 12, 18, 24, 36, 48, 54, . . .- A033845

20: 20, 40, 80,100,160,200,320, . . .

28: 28, 56,112,196,224,392,448, . . .

30: 30, 60, 90,120,150,180,240, . . .- A143207

42: 42, 84,126,168,252,294,336

...

Row 1 is T[1,1] = 1 and only has one term in the subsequence.

Row 7 is T[1,7] = 2*3*7; T[2,7] = 2^2*3*7; T[3,7] = 2*3^2*7; T[4,7] = 2^3*3*7; T[5,7] = 2^2*3^2*7, etc.

Examples

			a(14) = 80 and it is T[3,4] = 2^4*5. Its primitive progenitor is 20 = 2^2*5 and its equivalence class are the terms of row 4.
		

Links

• Frank M Jackson, Mathematica program

Crossrefs

Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.

Programs

Mathematica

(* See link above *)