A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset

1,2

Comments

Numbers such that none of the exponents in their prime factorization is of the form 3*m + 2.

Cohen (1962) proved that for a given number k >= 2 the asymptotic density of numbers whose exponents in their prime factorization are of the forms k*m or k*m + 1 only is zeta(k)/zeta(2). In this sequence k = 3, and therefore its asymptotic density is zeta(3)/zeta(2) = 6*zeta(3)/Pi^2 = 0.7307629694... (A253905).

Also, numbers k whose number of divisors, A000005(k), is not divisible by 3, i.e., complement of A059269.

Links

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

Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.

Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Example

6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 3*m + 2.
9 is not a term since 9 = 3^2 and 2 is of the form 3*m + 2.

Mathematica

Select[Range[100], Max[Mod[FactorInteger[#][[;; , 2]], 3]] < 2 &]

Crossrefs

Cf. A000005, A002117, A005117, A008834, A013661, A059269, A253905.

Union of A211337 and A211338.

Sequence in context: A326070 A337050 A304364 * A000452 A307295 A047587

Adjacent sequences: A336587 A336588 A336589 * A336591 A336592 A336593

Keyword

nonn

Author

Amiram Eldar, Jul 26 2020

Status

approved