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

1,1

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).

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. Tenebaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A017545, A201488 (coefficient in asymptotic formula), A332512, A332513, A332514, A332515, A332516.

Sequence in context: A239244 A267943 A066466 * A129293 A233512 A339675

Adjacent sequences: A332508 A332509 A332510 * A332512 A332513 A332514

Keyword

nonn

Author

Amiram Eldar, Feb 14 2020

Status

approved