A363216 Even powerful numbers that are not prime powers.
36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 484, 500, 576, 648, 676, 784, 800, 864, 900, 968, 972, 1000, 1152, 1156, 1296, 1352, 1372, 1444, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2116, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3136, 3200, 3364, 3456, 3528, 3600
Offset: 1
Keywords
Examples
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. a(2) = 72 = 2^3 * 3^2, a(3) = 100 = 2^2 * 5^2, etc.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
With[{nn = 3600}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[EvenQ[#], ! PrimePowerQ[#]] &] ] -
PARI
isok(k) = !(k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 27 2023
Formula
This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 0 }.
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
Comments