A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).

1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 8, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 4, 8, 16, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 4, 8, 1, 25, 1, 1, 9, 27, 1, 4, 1, 1, 1, 1, 4, 8, 16, 32, 1, 1, 1, 1, 4, 9, 36, 1, 1, 1, 1, 4, 8, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 4, 8, 16, 1, 49, 1, 25, 1, 1, 4

Offset: 1

Author

Michael De Vlieger, Feb 13 2025

Keywords

• nonn
• tabf
• easy

Comments

Intersection of row n of A027750 and A001694.

Examples

			D(1) = {1} = row 1 of this sequence since 1 | 1 is powerful.
D(2) = {1, 2}; of these, only 1 is powerful.
D(4) = {1, 2, 4}; of these, only 2 is not powerful, so row 4 = {1, 4}.
D(6) = {1, 2, 3, 6}; of these, only 1 is powerful.
D(8) = {1, 2, 4, 8}; of these, only 2 is not powerful, so row 4 = {1, 4, 8}.
D(12) = {1, 2, 3, 4, 6, 12}; of these, only {1, 4} are powerful.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, only {1, 4, 9, 36} are powerful, etc.
Table begins:
   n:  row n
  ----------------
   1:  1;
   2:  1;
   3:  1;
   4:  1, 4;
   5:  1;
   6:  1;
   7:  1;
   8:  1, 4, 8;
   9:  1, 9;
  10:  1;
  11:  1;
  12:  1, 4;
  13:  1;
  14:  1;
  15:  1;
  16:  1, 4, 8, 16;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11445 (rows n = 1..6000, flattened).

Crossrefs

Cf. A000005, A001694, A005361, A007947, A027750, A332785, A380819.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten

PARI

row(n) = select(x -> ispowerful(x), divisors(n)); \\ Amiram Eldar, May 02 2025

Formula

First term in row n is 1.

Row n does not contain squarefree factors of n, and also does not contain factors in A332785.

Length of row n = A005361(n) = tau(n/rad(n)), where tau = A000005 and rad = A007947.

For squarefree n, row n = {1}.

Let D(n) = row n of A027750. For prime p and m > 0, row p^m = D(p^m) \ {p}, since d = 1 and p = p^j, j > 1 are powerful, but primes are squarefree (and not powerful).