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A370786
Powerful numbers with an odd number of prime factors (counted with multiplicity).
4
8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608
OFFSET
1,1
COMMENTS
Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) - 1/zeta(3/2))/2 = 0.895230... .
The relative asymptotic density of this sequence within the powerful numbers is (1 - zeta(3)/(zeta(3/2)^2))/2 = 0.411930... .
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an odd number of prime factors (counted with multiplicity) within the s-full numbers is smaller than 1/2 when s is odd.
LINKS
MATHEMATICA
q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 1 &] && OddQ[Total[e]]]; Select[Range[2500], q]
PROG
(PARI) is(n) = {my(e = factor(n)[, 2]); n > 1 && vecmin(e) > 1 && vecsum(e)%2; }
CROSSREFS
Intersection of A001694 and A026424.
Complement of A370785 within A001694.
A370788 is a subsequence.
Sequence in context: A262675 A102834 A376171 * A377820 A116002 A339595
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 02 2024
STATUS
approved