A329229 Numbers that are the product of two odd prime powers with Euler phi-functions having solely a single 2 as a common prime factor.

15, 21, 33, 35, 39, 45, 51, 55, 57, 69, 75, 77, 87, 93, 95, 99, 111, 115, 119, 123, 129, 135, 141, 143, 147, 153, 155, 159, 161, 175, 177, 183, 187, 201, 203, 207, 209, 213, 215, 219, 225, 235, 237, 245, 249, 253, 261, 267, 287, 291, 295, 297, 299, 303, 309, 319

Offset: 1

Author

Gerold Brändli, Nov 08 2019

Keywords

• nonn

Comments

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).

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).

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).

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).

Trivially, the number of cyclic semiprimes of the form 3*p is infinite.

References

• Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Selbstverlag, Zürich, 2003, ISBN 3-9522917-0-6. See p. 15.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half The Amount of Residues, arXiv:1504.02757v2 [math.NT], 7 Feb 2016. See p. 10.
• Ki-Suk Lee, Miyeon Kwon and GiCheon Shin, Multiplicative Groups of Integers with Semi-primitive Roots Modulo n, Commun. Korean Math. Soc., Vol. 28, No. 1 (2013), pp. 71-77.
• Ki-Suk Lee, Ji-Eun Lee, Gerold Brändli and Tim Beyne, Galois Polynomials from Quotient Groups, Journal Chungcheong Math. Soc., Vol. 31, No. 3 (2018), pp. 309-319. See p. 311.

Crossrefs

Cf. A037074.

Programs

Maple

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)

Mathematica

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 *)

Extensions

More terms from Giovanni Resta, Dec 01 2019