A336592 Numbers k such that k/A008835(k) is cubefree, where A008835(k) is the largest 4th power dividing k.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset

1,2

Comments

Numbers such that none of the exponents in their prime factorization is of the form 4*m + 3.

Cohen (1963) proved that for a given number k > 2 the asymptotic density of numbers whose exponents in their prime factorization are not of the forms k*m - 1 is zeta(k)/zeta(k-1). In this sequence k = 4, and therefore its asymptotic density is zeta(4)/zeta(3) = Pi^4/(90*zeta(3)) = 0.9003926776...

Links

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

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

Example

6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 4*m + 3.
8 is not a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Mathematica

Select[Range[100], Max[Mod[FactorInteger[#][[;; , 2]], 4]] < 3 &]

Crossrefs

Cf. A002117, A004709, A008835, A013662.

Sequence in context: A219227 A007412 A270420 * A335275 A337052 A377020

Adjacent sequences: A336589 A336590 A336591 * A336593 A336594 A336595

Keyword

nonn

Author

Amiram Eldar, Jul 26 2020

Status

approved