A380033 Numbers that set records in A380032.

12, 36, 144, 576, 720, 900, 2880, 3600, 14400, 32400, 44100, 57600, 129600, 176400, 705600, 1587600, 2822400, 6350400, 11289600, 21344400, 25401600, 57153600, 85377600, 101606400, 192099600, 341510400, 768398400, 1366041600, 3073593600, 6915585600, 12294374400

Offset: 1

Author

Michael De Vlieger, Jan 11 2025

Keywords

• nonn

Comments

Proper subset of A364710 (intersection of A025487 and A126706).

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.

Conjecture 2: 36, 900, and 44100 are the only squares of primorials (in A061742) in the sequence.

Examples

			Let b(n) = A380032(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(5) = 2880 = 2^6 * 3^2 * 5, hence we write "6.2.1".
   n     a(n)  Exp.   b(a(n))
  --------------------------
   1      12   2.1        1   2*6
   2      36   2.2        2   2*18 = 3*12
   3     144   4.2        3   2*72 = 3*48 = 4*36
   4     576   6.2        4   2*288 = 3*192 = 4*144 = 8*72
   5     720   4.2.1      5   2*360 = 3*240 = 4*180 = 6*120 = 12*60
   6     900   2.2.2      6
   7    2880   6.2.1      7
   8    3600   4.2.2      9
   9   14400   6.2.2     12
  10   32400   4.4.2     13
  11   44100   2.2.2.2   14
  12   57600   8.2.2     15
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..66
• Michael De Vlieger, Prime power decomposition of a(n) n = 1..66, also including n = 67..141 (asterisked) that would follow if Conjecture 1 is true.
• Michael De Vlieger, List of (d, k/d), 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.

Crossrefs

Cf. A025487, A061742, A126706, A364710, A380032, A380034.

Programs

Mathematica

(* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Monitor[
  Do[k = s[[i]];
    If[# > r, r = #; Sow[k]] &@
      Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
        _?((m = GCD @@ {##};
          And[! MemberQ[{1, #2}, m],
          m == #1,
          ! Divisible[#1, rad[#2]]]) & @@ # &)], {i, nn}], i] ][[-1, 1]]