A377591 - OEIS

%I #6 Nov 13 2024 17:17:22

%S 225,675,1225,2025,3025,5929,6075,6125,8281,8575,14161,15125,18225,

%T 20449,30625,34969,41503,42875,43681,48841,54675,57967,60025,61009,

%U 64009,65219,75625,89401,99127,101761,104329,107653,116281,142129,152881,153125,162409,164025

%N 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.

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

%C Proper subset of A377590.

%H Michael De Vlieger, <a href="/A377591/b377591.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.18645.64480">Numbers k for which floor(log k / log lpf(k)) <= bigomega(k)</a>, 2024.

%e 36 is not in the sequence since 2^5 < 36, Omega(32) = 5, but Omega(36) = 4.

%e 72 is not in the sequence since 2^6 < 72, but Omega(72) = 5.

%e 225 is in the sequence since 3^4 < 225, Omega(81) = Omega(225) = 4.

%e 441 is not in the sequence since 3^5 < 441, Omega(243) = 5, but Omega(441) = 4, etc.

%t Select[With[{nn = 200000},

%t   Rest@ Select[

%t     Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],

%t     Not@*PrimePowerQ] ],

%t   Function[{n, k},

%t     NoneTrue[FactorInteger[n][[All, 1]],

%t       Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]

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

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Nov 02 2024