A377846 - OEIS

%I #9 Nov 10 2024 05:38:48

%S 1,4,8,9,16,25,27,32,36,49,64,72,81,100,108,121,125,128,144,169,196,

%T 200,225,243,256,288,289,324,343,361,392,400,441,484,500,512,529,576,

%U 625,675,676,729,784,800,841,900,961,968,972,1024,1089,1125,1152,1156,1225

%N Powerful numbers that are not divisible by the cubes of more than one distinct prime.

%C Subsequence of A377821 and first differs from it at n = 33: A377821(33) = 432 = 2^4 * 3^3 is not a term of this sequence.

%C Numbers whose prime factorization has exponents that are all larger than 1 and no more than one exponent is larger than 2.

%H Amiram Eldar, <a href="/A377846/b377846.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>.

%F Sum_{n>=1} 1/a(n) = (15/Pi^2) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .

%t q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, Length[e] == 1 || e[[-2]] == 2]; With[{max = 1300}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || q[#] &]]

%o (PARI) is(k) = if(k == 1, 1, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && (#e == 1 || e[#e - 1] == 2));

%Y Complement of A376936 within A001694.

%Y Subsequence of A377821.

%Y Subsequences: A143610, A377847.

%Y Cf. A082020.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Nov 09 2024