A379553 Numbers k in A376936 that set records in A379552.

216, 864, 3456, 7776, 31104, 124416, 279936, 497664, 972000, 1944000, 3888000, 7776000, 11664000, 15552000, 31104000, 34992000, 46656000, 62208000, 77760000, 97200000, 194400000, 291600000, 388800000, 777600000, 874800000, 1166400000, 1555200000, 3110400000, 3499200000

Offset: 1

Author

Michael De Vlieger, Dec 25 2024

Keywords

• nonn

Comments

Proper subset of A025487.

Examples

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Table below shows prime power decomposition of a(n), n = 1..12, writing only exponents in the "exp." column:
   n       a(n)   exp.   b(n)  (d,a(n)/d) with property Q
  -----------------------------------------------------------------
   1       216    3.3      1   (12,18)
   2       864    5.3      2   (18,48), (24,36)
   3      3456    7.3      3   (18,192), (36,96), (48,72)
   4      7776    5.5      4   (24,324), (48,162), (54,144), (72,108)
   5     31104    7.5      6
   6    124416    9.5      8
   7    279936    7.7      9
   8    497664   11.5     10
   9    972000    5.5.3   12
  10   1944000    6.5.3   14
  11   3888000    7.5.3   18
  12   7776000    8.5.3   20
See expanded table in links.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..212 (a(n) <= A002110(19).)
• Michael De Vlieger, Prime power decomposition of a(n), n = 1..212.

Crossrefs

Cf. A376936, A379552, A379554.

Programs

Mathematica

r = 0; nn = 10^9;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]; nn = Length[s];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
    rad[#1] == rad[#2],
    Mod[#1, #2] != 0,
    Mod[#2, #1] != 0] & @@ # &)], {i, nn}] ][[-1, 1]]