cp's OEIS Frontend

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

%I A332511 #18 Dec 11 2024 06:37:35
%S A332511 3,4,6,121,242,529,1058,2209,3481,4418,5041,6889,6962,10082,11449,
%T A332511 13778,14641,17161,22898,27889,29282,32041,34322,36481,51529,55778,
%U A332511 57121,63001,64082,69169,72962,96721,103058,114242,120409,126002,128881,138338,146689,175561
%N A332511 Numbers k such that phi(k) == 2 (mod 12), where phi is the Euler totient function (A000010).
%C A332511 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.
%C A332511 Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (1/2 + 1/(2*sqrt(2)))*sqrt(x)/log(x).
%H A332511 Amiram Eldar, <a href="/A332511/b332511.txt">Table of n, a(n) for n = 1..10000</a>
%H A332511 Joseph B. Dence and Thomas P. Dence, <a href="https://doi.org/10.1080/07468342.1995.11973716">A Surprise Regarding the Equation phi(x) = 2(6n + 1)</a>, The College Mathematics Journal, Vol. 26, No. 4 (1995), pp. 297-301.
%H A332511 Thomas Dence and Carl Pomerance, <a href="https://doi.org/10.1007/978-1-4757-4507-8_2">Euler's function in residue classes</a>, 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, <a href="https://math.dartmouth.edu/~carlp/PDF/paper116.pdf">alternative link</a>.
%e A332511 121 is a term since phi(121) = 110 == 2 (mod 12).
%t A332511 Select[Range[2*10^5], Mod[EulerPhi[#], 12] == 2 &]
%o A332511 (Magma) [k:k in [1..180000]| EulerPhi(k) mod 12 eq 2]; // _Marius A. Burtea_, Feb 14 2020
%Y A332511 Cf. A000010, A017545, A201488 (coefficient in asymptotic formula), A332512, A332513, A332514, A332515, A332516.
%K A332511 nonn
%O A332511 1,1
%A A332511 _Amiram Eldar_, Feb 14 2020