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

%I A379713 #20 Feb 14 2025 04:10:40
%S A379713 1,2,4,6,8,12,20,16,18,40,28,32,24,80,56,30,64,36,100,112,60,42,128,
%T A379713 48,160,196,90,84,66,256,54,200,224,120,126,132,78,512,72,320,392,150,
%U A379713 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
%N A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).
%C A379713 A permutation of the practical numbers.
%C A379713 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.
%C A379713 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.
%C A379713 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.
%C A379713 PP: Disjoint subsequence of A005153
%C A379713 --  -------------------------------
%C A379713 1:   1
%C A379713 2:   2,  4,  8, 16, 32, 64,128, . . .- A000079 with offset 1,1
%C A379713 6:   6, 12, 18, 24, 36, 48, 54, . . .- A033845
%C A379713 20: 20, 40, 80,100,160,200,320, . . .
%C A379713 28: 28, 56,112,196,224,392,448, . . .
%C A379713 30: 30, 60, 90,120,150,180,240, . . .- A143207
%C A379713 42: 42, 84,126,168,252,294,336
%C A379713 ...
%C A379713 Row 1 is T[1,1] = 1 and only has one term in the subsequence.
%C A379713 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.
%H A379713 Frank M Jackson, <a href="/A379713/a379713.txt">Mathematica program</a>
%e A379713 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.
%t A379713 (* See link above *)
%Y A379713 Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.
%K A379713 nonn,tabf
%O A379713 1,2
%A A379713 _Frank M Jackson_, Dec 30 2024