

A329229


Numbers that are the product of two odd prime powers with Euler phifunctions 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
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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 pessequence, 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 "semiprimitive root", if the order of x modulo s is phi(s)/2, where phi(s) is the Euler phifunction (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 3952291706, 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.
KiSuk Lee, Miyeon Kwon and GiCheon Shin, Multiplicative Groups of Integers with Semiprimitive Roots Modulo n, Commun. Korean Math. Soc. 28 (2013), No.1 pp.7177.
KiSuk Lee, JiEun 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



