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.
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, 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, 29, 2, 3, 5, 6, 10, 15, 30, 31, 2, 3, 11, 33, 2, 17, 34, 5, 7, 35, 2, 3, 6, 12, 18
Offset: 2
Examples
D(2) = {1, 2}; of these, only 2 is weak.
D(4) = {1, 2, 4}; of these, only 2 is weak.
D(6) = {1, 2, 3, 6}; of these, {2, 3, 6} are weak.
D(10) = {1, 2, 5, 10}; of these, {2, 5, 10} are weak.
D(12) = {1, 2, 3, 4, 6, 12}; of these, {2, 3, 6, 12} are weak.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, {2, 3, 6, 12, 18} are weak, etc.
Table begins:
n: row n
----------------
2: 2;
3: 3;
4: 2;
5: 5;
6: 2, 3, 6;
7: 7;
8: 2;
9: 3;
10: 2, 5, 10;
11: 11;
12: 2, 3, 6, 12;
13: 13;
14: 2, 7, 14;
15: 3, 5, 15;
...
Links
- Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).
Programs
-
Mathematica
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], ! Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten
Formula
Row 1 is empty since d = 1 is powerful (i.e., in A001694).
Let P(n) = row n of A027748 for n > 1. P(n) is a subset of row n.
Length of row n = A183093(n) = tau(n) = tau(n/rad(n)).
For prime p and m > 0, row p^m = {p}, since d = 1 and p = p^j, j > 1 are powerful.
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.
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