A091495 Odd squarefree numbers k such that k/phi(k) > 2, where phi is Euler's totient function.

105, 165, 195, 1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003, 3045, 3135, 3255, 3315, 3705, 3795, 3885, 3927, 4305, 4389, 4485, 4515, 4641, 4785, 4845, 4935, 5115, 5187, 5313, 5565, 5655, 5865, 6045, 6105, 6195, 6405, 6555, 6765, 7035, 7095, 7215

Offset: 1

Author

T. D. Noe, Jan 15 2004

Keywords

• nonn

Comments

Apparently the squarefree members of the sequence A036798. Note that 105, 165 and 195 are the only terms having 3 prime factors. Also note that all the numbers listed above have 3 as a factor. The smallest number of this form not divisible by 3 is 37182145 = 5*7*11*13*17*19*23.

From Amiram Eldar, Nov 21 2024: (Start)

If k is term and m is an odd squarefree number coprime to k, then k*m is also a term.

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 3, 58, 513, 5108, 52365, 523975, 5214831, 52103339, 521507571, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00521... . (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A000010, A036798, A056911, A118700.

Programs

Mathematica

lst={}; Do[f=FactorInteger[n]; s=Times@@Last/@f; If[s==1&&Times@@(1-1/(First/@f))<1/2, AppendTo[lst, n]], {n, 3, 10000, 2}]; lst
Select[Range[1,7301,2],SquareFreeQ[#]&&#/EulerPhi[#]>2&] (* Harvey P. Dale, Jul 10 2017 *)

PARI

is(k) = if(!(k%2), 0, my(f=factor(k)); issquarefree(f) && k / eulerphi(f) > 2); \\ Amiram Eldar, Nov 21 2024