A332511 Numbers k such that phi(k) == 2 (mod 12), where phi is the Euler totient function (A000010).

3, 4, 6, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 14641, 17161, 22898, 27889, 29282, 32041, 34322, 36481, 51529, 55778, 57121, 63001, 64082, 69169, 72962, 96721, 103058, 114242, 120409, 126002, 128881, 138338, 146689, 175561

Offset: 1

Author

Amiram Eldar, Feb 14 2020

Keywords

• nonn

Comments

Dence and Dence noted that the values of phi(k) congruent to 2 (mod 12) are sparse compared to the other possible even values. For example, for k <= 10^4 there only 10 values of phi(k) congruent to 2 (mod 12), compared to 6114, 1650, 511, 1233, and 476 values congruent to 0, 4, 6, 8, and 10 (mod 12), respectively. They proved that the asymptotic density of this sequence is 0 by showing that the only terms above 6 are of the form p^e and 2*p^e with p == 11 (mod 12) a prime and e even.

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (1/2 + 1/(2*sqrt(2)))*sqrt(x)/log(x).

Examples

			121 is a term since phi(121) = 110 == 2 (mod 12).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Joseph B. Dence and Thomas P. Dence, A Surprise Regarding the Equation phi(x) = 2(6n + 1), The College Mathematics Journal, Vol. 26, No. 4 (1995), pp. 297-301.
• 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, A017545, A201488 (coefficient in asymptotic formula), A332512, A332513, A332514, A332515, A332516.

Programs

Magma

[k:k in [1..180000]| EulerPhi(k) mod 12 eq 2]; // Marius A. Burtea, Feb 14 2020

Mathematica

Select[Range[2*10^5], Mod[EulerPhi[#], 12] == 2 &]