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A376170
Powerful numbers whose prime factorization has an even maximum exponent.
2
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 432, 441, 484, 529, 576, 625, 648, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2000, 2025, 2116, 2209
OFFSET
1,2
COMMENTS
Powerful numbers k such that A051903(k) is even.
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.
FORMULA
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).
MATHEMATICA
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]
PROG
(PARI) is(k) = {my(f = factor(k), e = f[, 2]); !(#e) || (ispowerful(f) && !(vecmax(e) % 2)); }
CROSSREFS
Complement of A376171 within A001694.
Intersection of A001694 and A368714.
A000290 \ {0} is a subsequence.
Cf. A051903.
Sequence in context: A370785 A069821 A331221 * A174452 A174902 A000290
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 13 2024
STATUS
approved