A363217 -id:A363217 - cp's OEIS frontend

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

Michael De Vlieger, May 21 2023

nonn

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

Subset of A001694, A126706, and A363101.

			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.

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

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

Cf. A002117, A013661, A013664, A082695.

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

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

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

A377591 Powerful numbers k that are not prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

225, 675, 1225, 2025, 3025, 5929, 6075, 6125, 8281, 8575, 14161, 15125, 18225, 20449, 30625, 34969, 41503, 42875, 43681, 48841, 54675, 57967, 60025, 61009, 64009, 65219, 75625, 89401, 99127, 101761, 104329, 107653, 116281, 142129, 152881, 153125, 162409, 164025

Offset: 1

Michael De Vlieger, Nov 02 2024

Terms are odd; proper subset of A363217, which is a proper subset of A286708, itself contained in A001694.

Proper subset of A377590.

			36 is not in the sequence since 2^5 < 36, Omega(32) = 5, but Omega(36) = 4.
72 is not in the sequence since 2^6 < 72, but Omega(72) = 5.
225 is in the sequence since 3^4 < 225, Omega(81) = Omega(225) = 4.
441 is not in the sequence since 3^5 < 441, Omega(243) = 5, but Omega(441) = 4, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024.

Cf. A001222, A001694, A007947, A286708, A363217, A376846, A377590.

Select[With[{nn = 200000},
  Rest@ Select[
    Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ] ],
  Function[{n, k},
    NoneTrue[FactorInteger[n][[All, 1]],
      Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]

cp's OEIS Frontend

A363216 Even powerful numbers that are not prime powers.

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