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With[{s = Merge[Map[#1 -> #2 & @@ # &, Position[ImageData[#], 0.]], Identity]}, Array[If[KeyExistsQ[s, #], Lookup[s, #], {0}] &, ImageDimensions[#][[-1]]] // Flatten] &@ Import["https://oeis.org/A349298/a349298.png"] (* Generate 1024 rows stored in the bitmap image, Michael De Vlieger, Dec 08 2021 *)

Michael De Vlieger, <a href="/A349298/a349298.png">1024-pixel bitmap</a> plotting (n, T(n,k)) in black, otherwise white including for rows containing 0.

Michael De Vlieger, <a href="/A349298/b349298.txt">Table of n, a(n) for n = 1..10635</a> (rows 1 <= n <= 600, flattened)

With[{nn = 45}, Table[If[Length[#] == 0, {0}, #] &@ Select[Array[# Boole[Xor[Or[Mod[#, 2] == Mod[n, 2] == 0, Mod[#, 3] == Mod[n, 3] == 0], GCD[n, #] != 1]] &, n], # > 0 &], {n, nn}]] // Flatten (* Michael De Vlieger, Dec 08 2021 *)

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allocated for Michael De VliegerPositions k in row n of triangles S(n,k) = T(n,k) = 0, where A054521 = S and A349297 = T, or 0 if there are no such k.

0, 0, 0, 0, 5, 0, 7, 0, 0, 5, 11, 0, 13, 7, 5, 10, 0, 17, 0, 19, 5, 15, 7, 14, 11, 23, 0, 5, 10, 15, 20, 25, 13, 0, 7, 21, 29, 5, 25, 31, 0, 11, 22, 17, 5, 7, 10, 14, 15, 20, 21, 25, 28, 30, 35, 0, 37, 19, 13, 26, 5, 15, 25, 35, 41, 7, 35, 43, 11, 33, 5, 10, 20, 25, 35, 40

1,5

Row n is a list of k for which A349297 NOR A054521 is true.

Row p > 3 for p prime has the single term p.

Nonzero terms in this sequence are of the form k*m > 1, where 3-smooth k > 1 in A003586 and 5-rough m > 1 in A007310, with m mod 6 = +/- 1.

Table T(n,k) for 1 <= n <= 16, replacing 0 with "." and 1 with "*", showing terms in row n of this sequence. Rows with no terms are replaced by 0:

1: .

2: . *

3: . . *

4: . * . *

5: . . . . 5

6: . * * * . *

7: . . . . . . 7

8: . * . * . * . *

9: . . * . . * . . *

10: . * . * 5 * . * . *

11: . . . . . . . . . . 11

12: . * * * . * . * * * . *

13: . . . . . . . . . . . . 13

14: . * . * . * 7 * . * . * . *

15: . . * . 5 * . . * 10 . * . . *

16: . * . * . * . * . * . * . * . *

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

n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Hence, row 5 = {5}, row 7 = {7}, row 11 = {11}, row 13 = {13}, row 14 = {7}, row 15 = {5, 10}, and all other rows 1 <= n <= 16 have no terms, thus are assigned 0 by definition.

Cf. A003586, A007310, A047229, A054521.

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Michael De Vlieger, Nov 13 2021

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allocated for Michael De Vlieger

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