A381823 Odd cubefree numbers that are not squarefree.

9, 25, 45, 49, 63, 75, 99, 117, 121, 147, 153, 169, 171, 175, 207, 225, 245, 261, 275, 279, 289, 315, 325, 333, 361, 363, 369, 387, 423, 425, 441, 475, 477, 495, 507, 525, 529, 531, 539, 549, 575, 585, 603, 605, 637, 639, 657, 693, 711, 725, 735, 747, 765, 775

Offset: 1

Author

Amiram Eldar, Mar 08 2025

Keywords

• nonn
• easy

Comments

Numbers whose prime factorization has only odd primes, exponents that are smaller than 3 and at least one exponent that equals 2.

Odd numbers k such that A051903(k) = A375039((k+1)/2) = 2.

The asymptotic density of this sequence is 4/(7*zeta(3)) - 2/(3*zeta(2)) = 0.070090906905338896329... .

In general, the asymptotic density of odd k-free numbers (numbers that are not divisible by a k-th power other than 1) that are not (k-1)-free, for k >= 2, is 2^(k-1)/((2^k-1) * zeta(k)) - 2^(k-2)/((2^(k-1)-1) * zeta(k-1)).

Links

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

Crossrefs

Intersection of A005408 and A067259.

Complement of A056911 within A381822.

Cf. A002117, A013661, A051903, A375039.

Subsequence of A048103.

Programs

Mathematica

Select[Range[1, 1000, 2], Max[FactorInteger[#][[;;, 2]]] == 2 &]

PARI

isok(k) = k % 2 && if(k == 1, 0, vecmax(factor(k)[, 2]) == 2);