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 A386433 #7 Jul 21 2025 13:15:18
%S A386433 108,648,972,1944,2700,3888,4500,8748,9000,13500,16200,17496,18000,
%T A386433 23328,24300,34992,36000,40500,45000,48600,52488,67500,69984,72000,
%U A386433 78732,81000,97200,112500,121500,132300,135000,139968,144000,145800,180000,209952,218700,220500
%N A386433 Achilles numbers with a primorial squarefree kernel that are not products of primorials.
%H A386433 Michael De Vlieger, <a href="/A386433/b386433.txt">Table of n, a(n) for n = 1..10000</a>
%F A386433 Let rad = A007947, omega = A001221, and P = A002110.
%F A386433 rad(a(n)) = P(omega(a(n))).
%F A386433 Intersection of A052486 and A056808 = A377854 \ A378002.
%e A386433 Table of n, a(n), and A053669(a(n)) for n = 1..12.
%e A386433 n a(n) A053669(a(n))
%e A386433 --------------------------------------------
%e A386433 1 108 = 2^2 * 3^3 5
%e A386433 2 648 = 2^3 * 3^4 5
%e A386433 3 972 = 2^2 * 3^5 5
%e A386433 4 1944 = 2^3 * 3^5 5
%e A386433 5 2700 = 2^2 * 3^3 * 5^2 7
%e A386433 6 3888 = 2^4 * 3^5 5
%e A386433 7 4500 = 2^2 * 3^2 * 5^3 7
%e A386433 8 8748 = 2^2 * 3^7 5
%e A386433 9 9000 = 2^3 * 3^2 * 5^3 7
%e A386433 10 13500 = 2^2 * 3^3 * 5^3 7
%e A386433 11 16200 = 2^3 * 3^4 * 5^2 7
%e A386433 12 17496 = 2^3 * 3^7 5
%e A386433 Let s = A052486.
%e A386433 The number 12 is not a term since it is not powerful (i.e., not in A001694).
%e A386433 The number 36, though powerful, is not a term since it is a perfect square.
%e A386433 s(1) = 72 is not in this sequence since rad(72) = P(2) = 6 and 72 = 2*6*6 = P(1)*P(2)*P(2).
%e A386433 s(2) = 108 = 3*6*6 is in the sequence since it is not a product of primorials.
%e A386433 The number 144, though powerful, is not a term because it is a perfect square.
%e A386433 s(3) = 200 is not a term because rad(200) = 10 = 2*5 is not also divisible by A053669(200) = 3.
%e A386433 s(4) = 288 is not in this sequence since rad(288) = P(2) = 6 and 288 = 2*2*2*6*6 = P(1)*P(1)*P(1)*P(2)*P(2), etc.
%t A386433 (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
%t A386433 Select[Union@ Flatten[f[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[;; , -1]] == 1, Max@ Differences[#2[[All, -1]] ] > 0] & @@ {#, FactorInteger[#]} &]
%Y A386433 Cf. A001221, A001597, A001694, A002110, A007947, A025487, A055932, A056808, A377854, A378002.
%K A386433 nonn
%O A386433 1,1
%A A386433 _Michael De Vlieger_, Jul 21 2025