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, 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, 301, 309, 321, 329, 341, 363, 381, 387, 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

Offset: 1

Author

Wolfdieter Lang, Jun 26 2020

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

Cf. A000010, A327922.

Programs

PARI

RRS(n) = select(x->gcd(n, x)==1, [1..n]); \\ A038566
RRSstar(n) = if (n<=2, [n-1], my(r=RRS(n)); Vec(r, #r/2)); \\ A333856
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

Formula

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.

Extensions

More terms from Michel Marcus, Sep 17 2023