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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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, ISBN 3-9522917-0-6, 2003 Zürich, p.15

LINKS

Table of n, a(n) for n=1..56.

Gerold Brändli, Tim Beyne, Modified Congruence Modulo n with Half The Amount of Residues, arXiv:1504.02757v2 [math.NT] 7 Feb 2016, p.10.

Ki-Suk Lee, Miyeon Kwon and GiCheon Shin, Multiplicative Groups of Integers with Semi-primitive Roots Modulo n, Commun. Korean Math. Soc. 28 (2013), No.1 pp.71-77.

Ki-Suk Lee, Ji-Eun Lee, Gerold Brändli and Tim Beyne, Galois Polynomials from Quotient Groups, Journal Chungcheong Math. Soc. 31 (2018), No.3, p. 311.

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 *)

CROSSREFS

Bisection of A062373 (odd indices).

Cf. A037074.

Sequence in context: A061346 A098905 A225375 * A146166 A024556 A046388

Adjacent sequences:  A329226 A329227 A329228 * A329230 A329231 A329232

KEYWORD

nonn

AUTHOR

Gerold Brändli, Nov 08 2019

EXTENSIONS

More terms from Giovanni Resta, Dec 01 2019

STATUS

approved

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Last modified September 18 03:39 EDT 2020. Contains 337164 sequences. (Running on oeis4.)