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A333857 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
1, 21, 33, 57, 63, 69, 77, 93, 99, 129, 133, 141, 147, 161, 171, 177, 189, 201, 207, 209, 213, 217, 231, 237, 249, 253, 279, 297 (list; graph; refs; listen; history; text; internal format)



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.

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).

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

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.


Table of n, a(n) for n=1..28.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.


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.


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

Sequence in context: A217263 A330441 A176945 * A070006 A189986 A190299

Adjacent sequences:  A333854 A333855 A333856 * A333858 A333859 A333860




Wolfdieter Lang, Jun 26 2020



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Last modified July 29 05:11 EDT 2021. Contains 346340 sequences. (Running on oeis4.)