cp's OEIS Frontend

A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.

%I A380819 #7 Feb 16 2025 23:02:26
%S A380819 2,3,2,5,2,3,6,7,2,3,2,5,10,11,2,3,6,12,13,2,7,14,3,5,15,2,17,2,3,6,
%T A380819 18,19,2,5,10,20,3,7,21,2,11,22,23,2,3,6,12,24,5,2,13,26,3,2,7,14,28,
%U A380819 29,2,3,5,6,10,15,30,31,2,3,11,33,2,17,34,5,7,35,2,3,6,12,18
%N A380819 Triangle read by rows where row n lists "weak" divisors d | n (i.e., d in A052485) such that rad(d)^2 does not divide d, where rad = A007947.
%C A380819 Intersection of row n of A027750 and A052485 for n > 1.
%H A380819 Michael De Vlieger, <a href="/A380819/b380819.txt">Table of n, a(n) for n = 2..11751</a> (rows n = 2..2000, flattened).
%F A380819 Row 1 is empty since d = 1 is powerful (i.e., in A001694).
%F A380819 Let P(n) = row n of A027748 for n > 1. P(n) is a subset of row n.
%F A380819 Length of row n = A183093(n) = tau(n) = tau(n/rad(n)).
%F A380819 For prime p and m > 0, row p^m = {p}, since d = 1 and p = p^j, j > 1 are powerful.
%F A380819 Let D(n) = row n of A027750. For squarefree composite n, row n = D(n) \ {1}, since d | n, d > 1, are squarefree for squarefree n.
%e A380819 D(2) = {1, 2}; of these, only 2 is weak.
%e A380819 D(4) = {1, 2, 4}; of these, only 2 is weak.
%e A380819 D(6) = {1, 2, 3, 6}; of these, {2, 3, 6} are weak.
%e A380819 D(10) = {1, 2, 5, 10}; of these, {2, 5, 10} are weak.
%e A380819 D(12) = {1, 2, 3, 4, 6, 12}; of these, {2, 3, 6, 12} are weak.
%e A380819 D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {2, 3, 6, 12, 18} are weak, etc.
%e A380819 Table begins:
%e A380819    n:  row n
%e A380819   ----------------
%e A380819    2:  2;
%e A380819    3:  3;
%e A380819    4:  2;
%e A380819    5:  5;
%e A380819    6:  2, 3, 6;
%e A380819    7:  7;
%e A380819    8:  2;
%e A380819    9:  3;
%e A380819   10:  2, 5, 10;
%e A380819   11:  11;
%e A380819   12:  2, 3, 6, 12;
%e A380819   13:  13;
%e A380819   14:  2, 7, 14;
%e A380819   15:  3, 5, 15;
%e A380819   ...
%t A380819 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], ! Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten
%Y A380819 Cf. A000005, A001694, A005361, A007947, A027750, A052485, A183093, A379545, A380672.
%K A380819 nonn,tabf,easy
%O A380819 2,1
%A A380819 _Michael De Vlieger_, Feb 13 2025