OFFSET
1,1
COMMENTS
Proper subset of A025487.
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.
EXAMPLE
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.
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]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Dec 25 2024
STATUS
approved