A380819 -id:A380819 - cp's OEIS frontend

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

Michael De Vlieger, Feb 13 2025

Intersection of row n of A027750 and A001694.

			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;
  ...

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

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

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

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

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).

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

Original entry on oeis.org

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

Michael De Vlieger, Feb 13 2025

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

			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;
  ...

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

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

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

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).

cp's OEIS Frontend

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

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