A381094 Triangle read by rows where row n contains k < n that are neither coprime to n nor have the same squarefree kernel as n, or 0 if there are no such k.

0, 0, 0, 0, 0, 2, 3, 4, 0, 6, 6, 2, 4, 5, 6, 8, 0, 2, 3, 4, 8, 9, 10, 0, 2, 4, 6, 7, 8, 10, 12, 3, 5, 6, 9, 10, 12, 6, 10, 12, 14, 0, 2, 3, 4, 8, 9, 10, 14, 15, 16, 0, 2, 4, 5, 6, 8, 12, 14, 15, 16, 18, 3, 6, 7, 9, 12, 14, 15, 18, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20

Offset: 1

Author

Michael De Vlieger, Feb 14 2025

Keywords

• nonn
• tabf
• easy

Comments

Let rad(k) = A007947(k), the squarefree kernel of k.

Let T(n) be row n of this sequence and let S(n) be row n of A133995.

T(n) contains numbers k < n such that k and n share at least one prime factor p, but not all distinct prime p | n.

T(n) is a superset of S(n), since S(n) does not contain any divisor d | n, while T(n) allows d | n such that rad(d) != rad(n).

Examples

			Table begins:
   n   row n
  ---------------------------
   1:  0;
   2:  0;
   3:  0;
   4:  0;
   5:  0;
   6:  2, 3, 4;
   7:  0;
   8:  6;
   9:  6;
  10:  2, 4, 5, 6, 8;
  11:  0;
  12:  2, 3, 4, 8, 9, 10;
  13:  0;
  14:  2, 4, 6, 7, 8, 10, 12;
  15:  3, 5, 6, 9, 10, 12;
  16:  6, 10, 12, 14;
From _Michael De Vlieger_, Mar 03 2025: (Start)
Row 10 is the union of {2, 4, 6, 8, 10} and {5, 10} without 10.
Row 12 is the union of {2, 4, 6, 8, 10, 12} and {3, 6, 9, 12} without {6, 12}.
Row 30 is the union of {2, 4, ..., 30}, {3, 6, ..., 30}, and {5, 10, ..., 30} without 30.
Row 84 is the union of {2, 4, ..., 84}, {3, 6, ..., 84}, and {7, 14, ..., 84} without {42, 84}, etc. (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11873 (rows n = 1..250, flattened)

Crossrefs

Cf. A007947, A121998, A133995, A369609, A381096.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

Formula

T(n) = { k < n : 1 < gcd(k,n), rad(k) != rad(n) }.

T(n) = S(n) \ { k : k | n, rad(k) = rad(n) }.

For prime p, T(p) = {}, but we write 0 to signify the empty set.

T(4) = 0, since k < 4 is either coprime to 4 or rad(k) = 2.

Let U(n) be row n of A121998 and let R(n) be row n of A369609. T(n) = U(n) \ R(n). - Michael De Vlieger, Mar 03 2025