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

1,1

Comments

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

Subset of A001694, A126706, and A363101.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Example

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.

Mathematica

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

Prog

(PARI) isok(k) = !(k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 27 2023

Crossrefs

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

Cf. A002117, A013661, A013664, A082695.

Sequence in context: A347960 A286708 A355462 * A363169 A114127 A322658

Adjacent sequences: A363213 A363214 A363215 * A363217 A363218 A363219

Keyword

nonn

Author

Michael De Vlieger, May 21 2023

Status

approved