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

1,6

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).

Links

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

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.

Example

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;

Mathematica

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

Crossrefs

Cf. A007947, A133995, A381096.

Sequence in context: A181578 A065332 A378212 * A265515 A336564 A129468

Adjacent sequences: A381090 A381092 A381093 * A381095 A381096 A381098

Keyword

nonn,tabf,easy,new

Author

Michael De Vlieger, Feb 14 2025

Status

approved