A382062 - OEIS

%I #9 Mar 14 2025 21:16:50

%S 1,8,27,32,72,108,125,128,200,216,243,343,392,432,500,512,648,675,864,

%T 968,1000,1125,1152,1323,1331,1352,1372,1728,1944,2000,2048,2187,2197,

%U 2312,2744,2888,3087,3125,3200,3267,3375,3456,4000,4232,4563,4913,5000,5324,5400

%N Powerful numbers whose number of divisors is divisible by their number of unitary divisors.

%C Powerful numbers k such that A034444(k) | A000005(k).

%C The primitive terms of A382061: if k is a term and m is a squarefree number that is coprime to k, then k*m is a term of A382061. The asymptotic density of A382061 can be calculated using the terms of this sequence (see A382061 for a formula).

%H Amiram Eldar, <a href="/A382062/b382062.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%e 27 = 3^3 is a term since it is powerful, A000005(27) = 4, A034444(27) = 2, and 2 | 4.

%e 72 = 2^3 * 3^2 is a term since it is powerful, A000005(72) = 12, A034444(72) = 4, and 4 | 12.

%t q[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, AllTrue[e, # > 1 &] && Divisible[Times @@ (e+1), 2^Length[e]]]; Select[Range[5400], # == 1 || q[#] &]

%o (PARI) isok(k) = if(k == 1, 1, my(f = factor(k)); vecmin(f[,2]) > 1 && !(numdiv(f) % (1<<omega(f))));

%Y Intersection of A001694 and A382061.

%Y Cf. A000005, A034444, A382064.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Mar 14 2025