A332513 Numbers k such that phi(k) == 4 (mod 12), where phi is the Euler totient function (A000010).

5, 8, 10, 12, 17, 29, 32, 34, 40, 41, 48, 53, 55, 58, 60, 75, 82, 85, 88, 89, 100, 101, 106, 110, 113, 115, 125, 128, 132, 136, 137, 145, 149, 150, 160, 170, 173, 178, 184, 187, 192, 197, 202, 204, 205, 226, 230, 232, 233, 235, 240, 250, 253, 257, 265, 269, 274

Offset: 1

Author

Amiram Eldar, Feb 14 2020

Keywords

• nonn

Comments

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ c1 * x/sqrt(log(x)), where c1 = (sqrt(2*sqrt(3))/(3*Pi)) * c3^(-1/2) * (2*c3 + c4) = 0.6109136202..., c3 = Product_{primes p == 2 (mod 3)} (1 + 1/(p^2-1)), and c4 = Product_{primes p == 2 (mod 3)} (1 - 1/(p+1)^2).

Examples

			17 is a term since phi(17) = 16 == 4 (mod 12).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Crossrefs

Cf. A000010, A017569, A175646, A332511, A332512, A332514, A332515, A332516.

Programs

Magma

[k:k in [1..300]| EulerPhi(k) mod 12 eq 4]; // Marius A. Burtea, Feb 14 2020

Mathematica

Select[Range[300], Mod[EulerPhi[#], 12] == 4 &]