A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 500, 512, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000

Offset

1,1

Comments

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.

Powerful numbers k such that A051903(k) is odd.

Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.

The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., 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 && OddQ[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 A376170 within A001694.

Intersection of A001694 and A376142.

Subsequence of A102834.

Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.

Cf. A051903, A082695, A376092.

Sequence in context: A354179 A262675 A102834 * A370786 A377820 A116002

Adjacent sequences: A376168 A376169 A376170 * A376172 A376173 A376174

Keyword

nonn,easy

Author

Amiram Eldar, Sep 13 2024

Status

approved