A376173 - OEIS

%I #11 Sep 13 2024 11:58:39

%S 8,27,32,125,128,216,243,343,432,512,648,864,1000,1331,1728,1944,2000,

%T 2048,2187,2197,2744,3125,3375,3456,4000,4913,5000,5488,5832,6859,

%U 6912,7776,8000,8192,9261,10125,10648,10976,12167,13824,15552,16000,16807,16875,17496

%N Numbers whose prime factorization has an odd minimum exponent that is larger than 1.

%C Numbers k such that A051904(k) is odd and larger than 1.

%C The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is not a term of this sequence.

%H Amiram Eldar, <a href="/A376173/b376173.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>=2} (-1)^k * s(k) = 0.2379998147971880759099..., 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 && OddQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[10000]

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

%Y Subsequence of A036966.

%Y Complement of A376172 within A001694.

%Y Subsequences: A030078, A062838 \ {1}.

%Y Cf. A051904.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Sep 13 2024