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

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, 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, 1, 1, 3, 11, 1, 2, 17, 1, 5, 7, 1, 2, 3, 4, 9, 1, 1, 2, 19

Offset

2,6

Comments

Row n lists terms in row n of A027750 that do not have the same squarefree kernel as does n.

Links

Michael De Vlieger, Table of n, a(n) for n = 2..11751 (rows n = 2..2000, flattened).

Formula

Row 1 is empty since rad(1) | rad(1).

The first term of row n is 1 for all n > 1.

n is not in row n since rad(n) = rad(n).

Length of row n = A183093(n) = tau(n) - tau(n/rad(n)).

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).

For prime p and m > 0, row p^m = {1}, since d | p^m, d > 1, are such that rad(d) = p.

For squarefree composite n, row n = D(n) \ {n} with length 2^(omega(k)-1).

Example

D(6) = {1, 2, 3, 6}; of these, {1, 2, 3} are such that rad(d) != rad(6).
D(10) = {1, 2, 5, 10}; of these, {1, 2, 5} are such that rad(d) != rad(10).
D(12) = {1, 2, 3, 4, 6, 12}; of these, {1, 2, 3, 4} are such that rad(d) != rad(12).
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.
Table begins:
   n:  row n
  ---------------
   2:  1;
   3:  1;
   4:  1;
   5:  1;
   6:  1, 2, 3;
   7:  1;
   8:  1;
   9:  1;
  10:  1, 2, 5;
  11:  1;
  12:  1, 2, 3, 4;
  13:  1;
  14:  1, 2, 7;
  15:  1, 3, 5;
  ...

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Select[Divisors[n], rad[#] != r &], {n, 2, 40}] // Flatten

Crossrefs

Cf. A000005, A005361, A007947, A027750, A183093, A284318, A380819.

Sequence in context: A276333 A347380 A380481 * A308638 A356320 A207572

Adjacent sequences: A380667 A380669 A380671 * A380673 A380674 A380675

Keyword

nonn,tabf,easy,new

Author

Michael De Vlieger, Feb 13 2025

Status

approved