A376170 - OEIS

%I #14 Sep 14 2024 06:52:29

%S 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,

%T 432,441,484,529,576,625,648,676,729,784,841,900,961,1024,1089,1156,

%U 1225,1296,1369,1444,1521,1600,1681,1728,1764,1849,1936,2000,2025,2116,2209

%N Powerful numbers whose prime factorization has an even maximum exponent.

%C Powerful numbers k such that A051903(k) is even.

%C Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is even. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is a term of this sequence.

%H Amiram Eldar, <a href="/A376170/b376170.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) = Sum_{k>=2} (-1)^k * s(k) = 1.65243302848832032223..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).

%t seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # == 1 || EvenQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]

%o (PARI) is(k) = {my(f = factor(k), e = f[,2]); !(#e) || (ispowerful(f) && !(vecmax(e) % 2));}

%Y Complement of A376171 within A001694.

%Y Intersection of A001694 and A368714.

%Y A000290 \ {0} is a subsequence.

%Y Cf. A051903.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Sep 13 2024