%I #17 May 28 2023 20:49:23

%S 36,72,100,108,144,196,200,216,288,324,392,400,432,484,500,576,648,

%T 676,784,800,864,900,968,972,1000,1152,1156,1296,1352,1372,1444,1568,

%U 1600,1728,1764,1800,1936,1944,2000,2116,2304,2312,2500,2592,2700,2704,2744,2888,2916,3136,3200,3364,3456,3528,3600

%N Even powerful numbers that are not prime powers.

%C This sequence is { A286708 INTERSECT A005843 } = { A001694 INTERSECT A363101 }.

%C Subset of A001694, A126706, and A363101.

%H Michael De Vlieger, <a href="/A363216/b363216.txt">Table of n, a(n) for n = 1..10000</a>

%F This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 0 }.

%F Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 = A082695 / 3 - 1/2 = 0.147865... . - _Amiram Eldar_, May 28 2023

%e a(1) = 36 = 2^2 * 3^2, the smallest even number with multiple distinct prime factors, all of which have multiplicity exceeding 1, so it is the first term.

%e a(2) = 72 = 2^3 * 3^2,

%e a(3) = 100 = 2^2 * 5^2, etc.

%t With[{nn = 3600}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[EvenQ[#], ! PrimePowerQ[#]] &] ]

%o (PARI) isok(k) = !(k%2) && ispowerful(k) && !isprimepower(k); \\ _Michel Marcus_, May 27 2023

%Y Cf. A001694, A062739, A126706, A286708, A363101, A363217.

%Y Cf. A002117, A013661, A013664, A082695.

%K nonn

%O 1,1

%A _Michael De Vlieger_, May 21 2023