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Positive odd numbers b with an unequal number of odd and even elements of the restricted residue system of the mod* congruence of Brändli and Beyne (numbers b ordered increasingly).
1

%I #13 Sep 17 2023 10:07:13

%S 1,21,33,57,63,69,77,93,99,129,133,141,147,161,171,177,189,201,207,

%T 209,213,217,231,237,249,253,279,297,301,309,321,329,341,363,381,387,

%U 393,399,413,417,423,437,441,453,469,473,483,489,497,501,513,517,531,537,539,553,567,573,581,589,597

%N Positive odd numbers b with an unequal number of odd and even elements of the restricted residue system of the mod* congruence of Brändli and Beyne (numbers b ordered increasingly).

%C For the modified congruence modulo n of Brändli and Beyne, called mod* n, see the link. See also the comments in A333856 for this reduced residue system mod* n, called RRS*(n), for n >= 1.

%C The odd members of RRS*(n) are denoted by RRS*odd(n), similarly, RRS*even(n) are the even elements of RRS*(n). E.g., RRS*odd(5) = {1} and RRS*even(5) = {2}. Therefore the odd number 5 can be called balanced in the reduced mod* system, because #RRS*odd(5) = 1 = #RRS*even(5).

%C All even numbers are unbalanced because RRS*(2*m) has only odd members, for m >= 1.

%C b = 1, with RRS*(1) = {0} is unbalanced, and for odd numbers b >= 3 to be balanced one needs integer phi(b)/4 because #RRS*(b) = phi(b)/2 (phi = A000010). The odd integers >= 3 with integer phi(b)/4 are given in A327922. The present sequence gives, for n >= 2, a proper subset of A327922 consisting of odd numbers b with an unequal number of odd and even elements (unbalanced odd b). Therefore, the condition phi(b)/4 integer for odd b is necessary but not sufficient for such odd b in the reduced mod* system.

%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:1504.02757 [math.NT], 2016.

%F This sequence gives the increasingly ordered positive odd integers b from A327922 such that the reduced residue system RRS*(b) does not have the same number of odd and even elements, for n >= 1, The odd number b is then called unbalanced.

%o (PARI) RRS(n) = select(x->gcd(n, x)==1, [1..n]); \\ A038566

%o RRSstar(n) = if (n<=2, [n-1], my(r=RRS(n)); Vec(r, #r/2)); \\ A333856

%o isok(k) = if ((k%2) && ((k==1) || denominator(eulerphi(k)/4)==1), my(v=RRSstar(k)); #select(x->((x%2) == 1), v) != #select(x->((x%2) == 0), v)); \\ _Michel Marcus_, Sep 17 2023

%Y Cf. A038566 (RRS(n)), A333856 (RRS*(n)).

%Y Cf. A000010, A327922.

%K nonn

%O 1,2

%A _Wolfdieter Lang_, Jun 26 2020

%E More terms from _Michel Marcus_, Sep 17 2023