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Numbers whose prime factorization has an even minimum exponent.
2

%I #10 Sep 13 2024 11:58:51

%S 1,4,9,16,25,36,49,64,72,81,100,108,121,144,169,196,200,225,256,288,

%T 289,324,361,392,400,441,484,500,529,576,625,675,676,729,784,800,841,

%U 900,961,968,972,1024,1089,1125,1152,1156,1225,1296,1323,1352,1369,1372,1444

%N Numbers whose prime factorization has an even minimum exponent.

%C Numbers k such that A051904(k) is even.

%C The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is a term of this sequence.

%H Amiram Eldar, <a href="/A376172/b376172.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) = 1 + Sum_{k>=1} (-1)^(k+1) * s(k) = 1.70559662202357112914..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).

%t seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # == 1 || EvenQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[2000]

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

%Y Subsequence of A001694.

%Y Complement of A376173 within A001694.

%Y Subsequences: A001248, A062503, A325240.

%Y Cf. A051904.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Sep 13 2024