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 A379553 #8 Dec 30 2024 17:09:44
%S A379553 216,864,3456,7776,31104,124416,279936,497664,972000,1944000,3888000,
%T A379553 7776000,11664000,15552000,31104000,34992000,46656000,62208000,
%U A379553 77760000,97200000,194400000,291600000,388800000,777600000,874800000,1166400000,1555200000,3110400000,3499200000
%N A379553 Numbers k in A376936 that set records in A379552.
%C A379553 Proper subset of A025487.
%H A379553 Michael De Vlieger, <a href="/A379553/b379553.txt">Table of n, a(n) for n = 1..212</a> (a(n) <= A002110(19).)
%H A379553 Michael De Vlieger, <a href="/A379553/a379553.txt">Prime power decomposition of a(n)</a>, n = 1..212.
%e A379553 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:
%e A379553 n a(n) exp. b(n) (d,a(n)/d) with property Q
%e A379553 -----------------------------------------------------------------
%e A379553 1 216 3.3 1 (12,18)
%e A379553 2 864 5.3 2 (18,48), (24,36)
%e A379553 3 3456 7.3 3 (18,192), (36,96), (48,72)
%e A379553 4 7776 5.5 4 (24,324), (48,162), (54,144), (72,108)
%e A379553 5 31104 7.5 6
%e A379553 6 124416 9.5 8
%e A379553 7 279936 7.7 9
%e A379553 8 497664 11.5 10
%e A379553 9 972000 5.5.3 12
%e A379553 10 1944000 6.5.3 14
%e A379553 11 3888000 7.5.3 18
%e A379553 12 7776000 8.5.3 20
%e A379553 See expanded table in links.
%t A379553 r = 0; nn = 10^9;
%t A379553 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379553 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];
%t A379553 Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
%t A379553 Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A379553 _?(And[1 < GCD @@ {##},
%t A379553 rad[#1] == rad[#2],
%t A379553 Mod[#1, #2] != 0,
%t A379553 Mod[#2, #1] != 0] & @@ # &)], {i, nn}] ][[-1, 1]]
%Y A379553 Cf. A376936, A379552, A379554.
%K A379553 nonn
%O A379553 1,1
%A A379553 _Michael De Vlieger_, Dec 25 2024