This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379545 #31 May 02 2025 03:17:02
%S A379545 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,
%T A379545 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,
%U A379545 1,1,1,4,1,9,1,1,1,4,8,16,1,49,1,25,1,1,4
%N A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).
%C A379545 Intersection of row n of A027750 and A001694.
%H A379545 Michael De Vlieger, <a href="/A379545/b379545.txt">Table of n, a(n) for n = 1..11445</a> (rows n = 1..6000, flattened).
%F A379545 First term in row n is 1.
%F A379545 Row n does not contain squarefree factors of n, and also does not contain factors in A332785.
%F A379545 Length of row n = A005361(n) = tau(n/rad(n)), where tau = A000005 and rad = A007947.
%F A379545 For squarefree n, row n = {1}.
%F A379545 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).
%e A379545 D(1) = {1} = row 1 of this sequence since 1 | 1 is powerful.
%e A379545 D(2) = {1, 2}; of these, only 1 is powerful.
%e A379545 D(4) = {1, 2, 4}; of these, only 2 is not powerful, so row 4 = {1, 4}.
%e A379545 D(6) = {1, 2, 3, 6}; of these, only 1 is powerful.
%e A379545 D(8) = {1, 2, 4, 8}; of these, only 2 is not powerful, so row 4 = {1, 4, 8}.
%e A379545 D(12) = {1, 2, 3, 4, 6, 12}; of these, only {1, 4} are powerful.
%e A379545 D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, only {1, 4, 9, 36} are powerful, etc.
%e A379545 Table begins:
%e A379545 n: row n
%e A379545 ----------------
%e A379545 1: 1;
%e A379545 2: 1;
%e A379545 3: 1;
%e A379545 4: 1, 4;
%e A379545 5: 1;
%e A379545 6: 1;
%e A379545 7: 1;
%e A379545 8: 1, 4, 8;
%e A379545 9: 1, 9;
%e A379545 10: 1;
%e A379545 11: 1;
%e A379545 12: 1, 4;
%e A379545 13: 1;
%e A379545 14: 1;
%e A379545 15: 1;
%e A379545 16: 1, 4, 8, 16;
%e A379545 ...
%t A379545 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten
%o A379545 (PARI) row(n) = select(x -> ispowerful(x), divisors(n)); \\ _Amiram Eldar_, May 02 2025
%Y A379545 Cf. A000005, A001694, A005361, A007947, A027750, A332785, A380819.
%K A379545 nonn,tabf,easy
%O A379545 1,5
%A A379545 _Michael De Vlieger_, Feb 13 2025