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.

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

Author

Michael De Vlieger, Nov 02 2024

Keywords

• nonn
• easy

Comments

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

Proper subset of A377590.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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[#]} &]