A344085 - OEIS

A344085

Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Differs from A339195 in having 77 before 66.

LINKS

Table of n, a(n) for n=1..64.

FORMULA

From Michael De Vlieger, Jan 03 2025: (Start)

a(n) = A019565(A187769(n-1)).

As an irregular triangle T(n,k), where row 0 = {1}:

For n > 1, k = 1..2^(n-1).

For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.) (End)

EXAMPLE

Triangle begins:

3 6

5 10 15 30

7 14 21 35 42 70 105 210

From Michael De Vlieger, Jan 03 2025: (Start)

Table of a(n) for n = 1..32, demonstrating relationship of this sequence with s = A187769:

<-factors <-factors

n-1 a(n) 2 3 5 7 s(n) | n-1 a(n) 2 3 5 7 11 s(n)

-------------------------|----------------------------

0 1 . 0 | 16 11 . . . . x 16

1 2 x 1 | 17 22 x . . . x 17

2 3 . x 2 | 18 33 . x . . x 18

3 6 x x 3 | 19 55 . . x . x 20

4 5 . . x 4 | 20 77 . . . x x 24

5 10 x . x 5 | 21 66 x x . . x 19

6 15 . x x 6 | 22 110 x . x . x 21

7 30 x x x 7 | 23 154 x . . x x 22

8 7 . . . x 8 | 24 165 . x x . x 25

9 14 x . . x 9 | 25 231 . x . x x 26

10 21 . x . x 10 | 26 385 . . x x x 28

11 35 . . x x 12 | 27 330 x x x . x 23

12 42 x x . x 11 | 28 462 x x . x x 27

13 70 x . x x 13 | 29 770 x . x x x 29

14 105 . x x x 14 | 30 1155 . x x x x 30

15 210 x x x x 15 | 31 2310 x x x x x 31

-------------------------|----------------------------

1 2 4 8 s(n) | 1 2 4 8 16 s(n)

bits-> bits-> (End)

MATHEMATICA

nn=4;

GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]], SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1, 1]]]<=nn&], PrimeOmega], FactorInteger[#][[-1, 1]]&]

(* Second, faster program: *)

With[{m = 8},

Map[SortBy[#, PrimeNu] &,

TakeList[

Array[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,

Reverse@ IntegerDigits[#, 2]] &, 2^m, 0],

{1}~Join~(2^Range[0, m - 1] ) ] ] ] // Flatten (* Michael De Vlieger, Jan 03 2025 *)

CROSSREFS

Row lengths are A000079.

Grouping by greatest prime factor only gives A339195.

Row sums are 1 and A339360.

Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.