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%I #44 Oct 19 2021 02:51:45
%S 15,21,33,35,39,45,51,55,57,69,75,77,87,93,95,99,111,115,119,123,129,
%T 135,141,143,147,153,155,159,161,175,177,183,187,201,203,207,209,213,
%U 215,219,225,235,237,245,249,253,261,267,287,291,295,297,299,303,309,319
%N Numbers that are the product of two odd prime powers with Euler phi-functions having solely a single 2 as a common prime factor.
%C Numbers p^j*q^k, denoted "cyclic semiprimes", such that gcd(phi(p^j), phi(q^k)) = 2, p and q odd primes, j and k positive integers (Brändli and Beyne, 2016, def.4 and Lee et al., 2013, theo.1).
%C The products of twin primes (A037074), and odd composite numbers with a single pes-sequence, i.e. parameter B = 1, are a subset of this sequence (Schick 2003, eq.1.6.2).
%C Any element x in Zs* is said to be a "semi-primitive root", if the order of x modulo s is phi(s)/2, where phi(s) is the Euler phi-function (Lee 2013, def.1).
%C If s is a cyclic semiprime, x is a generating element and k an integer, then the following reduced modulus denoted mod* returns all elements of Zs* in the interval ]0,s/2[, with mod* defined by x^k mod* s = min(+-x^k mod s) (Lee et al., 2018, def.2.3).
%C Trivially, the number of cyclic semiprimes of the form 3*p is infinite.
%D Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Selbstverlag, Zürich, 2003, ISBN 3-9522917-0-6. See p. 15.
%H Amiram Eldar, <a href="/A329229/b329229.txt">Table of n, a(n) for n = 1..10000</a>
%H Gerold Brändli and Tim Beyne, <a href="https://arxiv.org/abs/1504.02757">Modified Congruence Modulo n with Half The Amount of Residues</a>, arXiv:1504.02757v2 [math.NT], 7 Feb 2016. See p. 10.
%H Ki-Suk Lee, Miyeon Kwon and GiCheon Shin, <a href="https://dx.doi.org/10.4134/CKMS.2013.28.1.071">Multiplicative Groups of Integers with Semi-primitive Roots Modulo n</a>, Commun. Korean Math. Soc., Vol. 28, No. 1 (2013), pp. 71-77.
%H Ki-Suk Lee, Ji-Eun Lee, Gerold Brändli and Tim Beyne, <a href="http://dx.doi.org/10.14403/jcms.2018.31.1.309">Galois Polynomials from Quotient Groups</a>, Journal Chungcheong Math. Soc., Vol. 31, No. 3 (2018), pp. 309-319. See p. 311.
%p with(NumberTheory, Totient, PrimitiveRoot, Divisors, tau, phi, lambda); K := {}; for i from 3 by 2 to 100 do for j from i+2 by 2 to 100 do if numelems(ifactors(i*j)[2]) = 2 and gcd(phi(i), phi(j)) = 2 and gcd(i, j) = 1 then K := K union {i*j} end if end do end do; print(K)
%t Select[Range[5, 320, 2], (f = FactorInteger[#]; Length[f] == 2 && GCD[ EulerPhi[ f[[1, 1]]^f[[1, 2]]], EulerPhi[f[[2, 1]]^f[[2, 2]]]] == 2) &] (* _Giovanni Resta_, Dec 01 2019 *)
%Y Bisection of A062373 (odd indices).
%Y Cf. A037074.
%K nonn
%O 1,1
%A _Gerold Brändli_, Nov 08 2019
%E More terms from _Giovanni Resta_, Dec 01 2019
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