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 A380033 #6 Jan 15 2025 08:43:21
%S A380033 12,36,144,576,720,900,2880,3600,14400,32400,44100,57600,129600,
%T A380033 176400,705600,1587600,2822400,6350400,11289600,21344400,25401600,
%U A380033 57153600,85377600,101606400,192099600,341510400,768398400,1366041600,3073593600,6915585600,12294374400
%N A380033 Numbers that set records in A380032.
%C A380033 Proper subset of A364710 (intersection of A025487 and A126706).
%C A380033 Conjecture 1: Almost all numbers in this sequence are powerful squares. Only 12, 720, and 2880 are not powerful. Thereby this sequence is a proper subset of A368682 (intersection of A025487 and A131605, the latter a subset of A001597 and A286708), in turn a subset of A364710.
%C A380033 Conjecture 2: 36, 900, and 44100 are the only squares of primorials (in A061742) in the sequence.
%H A380033 Michael De Vlieger, <a href="/A380033/b380033.txt">Table of n, a(n) for n = 1..66</a>
%H A380033 Michael De Vlieger, <a href="/A380033/a380033.txt">Prime power decomposition of a(n)</a> n = 1..66, also including n = 67..141 (asterisked) that would follow if Conjecture 1 is true.
%H A380033 Michael De Vlieger, <a href="/A380033/a380033.png">List of (d, k/d)</a>, d < k/d, k = a(n), n = 1..24, d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d.
%e A380033 Let b(n) = A380032(n).
%e A380033 Table showing exponents of prime power factors of a(n) for n = 1..12.
%e A380033 Example: a(5) = 2880 = 2^6 * 3^2 * 5, hence we write "6.2.1".
%e A380033 n a(n) Exp. b(a(n))
%e A380033 --------------------------
%e A380033 1 12 2.1 1 2*6
%e A380033 2 36 2.2 2 2*18 = 3*12
%e A380033 3 144 4.2 3 2*72 = 3*48 = 4*36
%e A380033 4 576 6.2 4 2*288 = 3*192 = 4*144 = 8*72
%e A380033 5 720 4.2.1 5 2*360 = 3*240 = 4*180 = 6*120 = 12*60
%e A380033 6 900 2.2.2 6
%e A380033 7 2880 6.2.1 7
%e A380033 8 3600 4.2.2 9
%e A380033 9 14400 6.2.2 12
%e A380033 10 32400 4.4.2 13
%e A380033 11 44100 2.2.2.2 14
%e A380033 12 57600 8.2.2 15
%t A380033 (* Load function f at A025487 *)
%t A380033 r = 0;
%t A380033 s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
%t A380033 nn = Length[s]; Print[nn];
%t A380033 Reap[Monitor[
%t A380033 Do[k = s[[i]];
%t A380033 If[# > r, r = #; Sow[k]] &@
%t A380033 Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A380033 _?((m = GCD @@ {##};
%t A380033 And[! MemberQ[{1, #2}, m],
%t A380033 m == #1,
%t A380033 ! Divisible[#1, rad[#2]]]) & @@ # &)], {i, nn}], i] ][[-1, 1]]
%Y A380033 Cf. A025487, A061742, A126706, A364710, A380032, A380034.
%K A380033 nonn
%O A380033 1,1
%A A380033 _Michael De Vlieger_, Jan 11 2025