cp's OEIS Frontend

A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.

%I A380672 #25 Feb 16 2025 23:01:56
%S A380672 1,1,1,1,1,2,3,1,1,1,1,2,5,1,1,2,3,4,1,1,2,7,1,3,5,1,1,1,2,3,9,1,1,2,
%T A380672 4,5,1,3,7,1,2,11,1,1,2,3,4,8,1,1,2,13,1,1,2,4,7,1,1,2,3,5,6,10,15,1,
%U A380672 1,1,3,11,1,2,17,1,5,7,1,2,3,4,9,1,1,2,19
%N A380672 Triangle read by rows where row n lists divisors d | n such that rad(d) != rad(n), where rad = A007947.
%C A380672 Row n lists terms in row n of A027750 that do not have the same squarefree kernel as does n.
%H A380672 Michael De Vlieger, <a href="/A380672/b380672.txt">Table of n, a(n) for n = 2..11751</a> (rows n = 2..2000, flattened).
%F A380672 Row 1 is empty since rad(1) | rad(1).
%F A380672 The first term of row n is 1 for all n > 1.
%F A380672 n is not in row n since rad(n) = rad(n).
%F A380672 Length of row n = A183093(n) = tau(n) - tau(n/rad(n)).
%F A380672 Let S(n) = row n of A284318 and let D(n) = row n of A027750. Then row n of this sequence is D(n) \ S(n).
%F A380672 For prime p and m > 0, row p^m = {1}, since d | p^m, d > 1, are such that rad(d) = p.
%F A380672 For squarefree composite n, row n = D(n) \ {n} with length 2^(omega(k)-1).
%e A380672 D(6) = {1, 2, 3, 6}; of these, {1, 2, 3} are such that rad(d) != rad(6).
%e A380672 D(10) = {1, 2, 5, 10}; of these, {1, 2, 5} are such that rad(d) != rad(10).
%e A380672 D(12) = {1, 2, 3, 4, 6, 12}; of these, {1, 2, 3, 4} are such that rad(d) != rad(12).
%e A380672 D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {1, 2, 3, 4, 9} are such that rad(d) != rad(36), etc.
%e A380672 Table begins:
%e A380672    n:  row n
%e A380672   ---------------
%e A380672    2:  1;
%e A380672    3:  1;
%e A380672    4:  1;
%e A380672    5:  1;
%e A380672    6:  1, 2, 3;
%e A380672    7:  1;
%e A380672    8:  1;
%e A380672    9:  1;
%e A380672   10:  1, 2, 5;
%e A380672   11:  1;
%e A380672   12:  1, 2, 3, 4;
%e A380672   13:  1;
%e A380672   14:  1, 2, 7;
%e A380672   15:  1, 3, 5;
%e A380672   ...
%t A380672 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Divisors[n], rad[#] != r &], {n, 2, 40}] // Flatten
%Y A380672 Cf. A000005, A005361, A007947, A027750, A183093, A284318, A380819.
%K A380672 nonn,tabf,easy
%O A380672 2,6
%A A380672 _Michael De Vlieger_, Feb 13 2025