A349317 - OEIS

A349317

Triangle T(n,k): T(n,k) = 1 if gcd(n, k) > 1, else 0.

%I #9 Dec 09 2021 01:01:03

%S 0,0,1,0,0,1,0,1,0,1,0,0,0,0,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0,1,0,1,

%T 0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1,

%U 1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1

%N Triangle T(n,k): T(n,k) = 1 if gcd(n, k) > 1, else 0.

%C Characteristic function of k in the cototient of n, i.e., of A169582: a(A169582(n)) = 1; a(A169581(n)) = 0.

%H Michael De Vlieger, <a href="/A349317/b349317.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)

%H Michael De Vlieger, <a href="/A349317/a349317.png">Bitmap of 256 rows</a> enlarged 4X, with 1's in black and 0's in white.

%H Michael De Vlieger, <a href="/A349317/a349317_1.png">Bitmap of 4096 rows</a> with 1's in black and 0's in white.

%F Row sums are in A051953 = n - A000010(n).

%F A067392 = Sum of k*T(n,k).

%F A066570 = Product of k in row n such that T(n,k)=1

%F Inverse of A054521 = S(n,k); T(n,k) = 1 - S(n,k).

%e Table T(n,k) for 1 <= n <= 16, replacing 0 with "." for clarity:

%e 1: .

%e 2: . 1

%e 3: . . 1

%e 4: . 1 . 1

%e 5: . . . . 1

%e 6: . 1 1 1 . 1

%e 7: . . . . . . 1

%e 8: . 1 . 1 . 1 . 1

%e 9: . . 1 . . 1 . . 1

%e 10: . 1 . 1 1 1 . 1 . 1

%e 11: . . . . . . . . . . 1

%e 12: . 1 1 1 . 1 . 1 1 1 . 1

%e 13: . . . . . . . . . . . . 1

%e 14: . 1 . 1 . 1 1 1 . 1 . 1 . 1

%e 15: . . 1 . 1 1 . . 1 1 . 1 . . 1

%e 16: . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1

%e ---------------------------------------------------

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

%t Table[Boole[GCD[n, k] > 1], {n, 13}, {k, n}] // Flatten

%t (* Generate up to 4096 rows = 8390656 terms from data stored in the bitmap: *)

%t With[{rows = 150, s = ImageData[Import["https://oeis.org/A349317/a349317_1.png"]]}, Array[s[[#, 1 ;; #]] /. {0. -> 1, 1. -> 0} &, rows]] (* _Michael De Vlieger_, Dec 08 2021 *)

%Y Cf. A000010, A051953, A054521, A067392, A066570.

%K nonn,easy,tabl

%O 1

%A _Michael De Vlieger_, Nov 14 2021