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A377818
Powerful numbers that have a single even exponent in their prime factorization.
3
4, 9, 16, 25, 49, 64, 72, 81, 108, 121, 169, 200, 256, 288, 289, 361, 392, 432, 500, 529, 625, 648, 675, 729, 800, 841, 961, 968, 972, 1024, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 2000, 2209, 2312, 2401, 2592, 2809, 2888, 3087, 3200, 3267, 3481
OFFSET
1,1
COMMENTS
Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is a cubefull exponentially odd number (A335988) and p is a prime that does not divide m.
Powerful numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * Sum_{p prime} (p/(p^3-p+1)) = 0.61399274770712398109... .
MATHEMATICA
With[{max = 3500}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]]
PROG
(PARI) is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) == 1);
CROSSREFS
Intersection of A001694 and A377816.
Subsequence of A377819.
Sequence in context: A065741 A188061 A279456 * A069560 A075494 A063735
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Nov 09 2024
STATUS
approved