A381094 - OEIS

%I #9 Feb 19 2025 11:56:04

%S 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,

%T 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,

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

%N 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.

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

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

%C 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.

%C 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).

%H Michael De Vlieger, <a href="/A381094/b381094.txt">Table of n, a(n) for n = 1..11873</a> (rows n = 1..250, flattened)

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

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

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

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

%e Table begins:

%e    n   row n

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

%e    1:  0;

%e    2:  0;

%e    3:  0;

%e    4:  0;

%e    5:  0;

%e    6:  2, 3, 4;

%e    7:  0;

%e    8:  6;

%e    9:  6;

%e   10:  2, 4, 5, 6, 8;

%e   11:  0;

%e   12:  2, 3, 4, 8, 9, 10;

%e   13:  0;

%e   14:  2, 4, 6, 7, 8, 10, 12;

%e   15:  3, 5, 6, 9, 10, 12;

%e   16:  6, 10, 12, 14;

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

%Y Cf. A007947, A133995, A381096.

%K nonn,tabf,easy,new

%O 1,6

%A _Michael De Vlieger_, Feb 14 2025