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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.

%I A381094 #12 Mar 03 2025 13:22:27
%S A381094 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 A381094 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 A381094 18,3,6,7,9,12,14,15,18,2,4,6,8,10,11,12,14,16,18,20
%N 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.
%C A381094 Let rad(k) = A007947(k), the squarefree kernel of k.
%C A381094 Let T(n) be row n of this sequence and let S(n) be row n of A133995.
%C A381094 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 A381094 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 A381094 Michael De Vlieger, <a href="/A381094/b381094.txt">Table of n, a(n) for n = 1..11873</a> (rows n = 1..250, flattened)
%F A381094 T(n) = { k < n : 1 < gcd(k,n), rad(k) != rad(n) }.
%F A381094 T(n) = S(n) \ { k : k | n, rad(k) = rad(n) }.
%F A381094 For prime p, T(p) = {}, but we write 0 to signify the empty set.
%F A381094 T(4) = 0, since k < 4 is either coprime to 4 or rad(k) = 2.
%F A381094 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
%e A381094 Table begins:
%e A381094    n   row n
%e A381094   ---------------------------
%e A381094    1:  0;
%e A381094    2:  0;
%e A381094    3:  0;
%e A381094    4:  0;
%e A381094    5:  0;
%e A381094    6:  2, 3, 4;
%e A381094    7:  0;
%e A381094    8:  6;
%e A381094    9:  6;
%e A381094   10:  2, 4, 5, 6, 8;
%e A381094   11:  0;
%e A381094   12:  2, 3, 4, 8, 9, 10;
%e A381094   13:  0;
%e A381094   14:  2, 4, 6, 7, 8, 10, 12;
%e A381094   15:  3, 5, 6, 9, 10, 12;
%e A381094   16:  6, 10, 12, 14;
%e A381094 From _Michael De Vlieger_, Mar 03 2025: (Start)
%e A381094 Row 10 is the union of {2, 4, 6, 8, 10} and {5, 10} without 10.
%e A381094 Row 12 is the union of {2, 4, 6, 8, 10, 12} and {3, 6, 9, 12} without {6, 12}.
%e A381094 Row 30 is the union of {2, 4, ..., 30}, {3, 6, ..., 30}, and {5, 10, ..., 30} without 30.
%e A381094 Row 84 is the union of {2, 4, ..., 84}, {3, 6, ..., 84}, and {7, 14, ..., 84} without {42, 84}, etc. (End)
%t A381094 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]
%Y A381094 Cf. A007947, A121998, A133995, A369609, A381096.
%K A381094 nonn,tabf,easy
%O A381094 1,6
%A A381094 _Michael De Vlieger_, Feb 14 2025