A216319 -id:A216319 - cp's OEIS frontend

A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1.

1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 5, 3, 1, 5, 3, 7, 1, 7, 5, 1, 7, 3, 9, 1, 7, 9, 3, 5, 1, 5, 7, 11, 1, 9, 5, 11, 3, 7, 1, 9, 11, 3, 5, 13, 1, 13, 11, 7, 1, 11, 13, 9, 7, 3, 5, 15, 1, 11, 7, 5, 15, 3, 13, 9, 1, 7, 5, 13, 11, 17, 1, 13, 15, 11, 17, 7, 3, 5, 9, 1, 13, 17, 9, 11, 3, 7, 19, 1, 17, 19, 13, 5, 11, 1, 15, 9, 19, 5, 17, 3, 13, 7, 21

Offset: 1

Wolfdieter Lang, Oct 20 2020

The length of row n is A055034(n), called here delta(n), for n >= 1.
For the modified modular equivalence relation Modd n see a comment in A203571, and the W. Lang link, Definition 4. p. 25. For Modd(a, n) one has to consider the parity of floor(a/n). If it is even then Modd(a, n) = mod(a, n), otherwise it is mod(-a, n).
The rows of A216319 are the smallest positive restricted residue system mod n with only odd members (RRSodd(n)). This is not a group mod n, but a group Modd n, called here G(rho(n)). This group is isomorphic to the Galois group Gal(Q(rho(n))/Q), where the algebraic number of degree delta(n) is rho(n) = 2*cos(Pi/n), for n >= 1. See A187360 for the minimal polynomials of rho(n), called C(n, x).

			The irregular triangle T(n, k) begins:
n\k 1  2  3  4  5  6  7  8 9 ...
1:  1
2:  1
3:  1
4:  1  3
5:  1  3
6:  1  5
7:  1  5  3
8:  1  5  3  7
9:  1  7  5
10: 1  7  3  9
11: 1  7  9  3  5
12: 1  5  7 11
13: 1  9  5 11  3  7
14: 1  9 11  3  5 13
15: 1 13 11  7
16: 1 11 13  9  7  3  5 15
17: 1 11  7  5 15  3 13  9
18: 1  7  5 13 11 17
19: 1 13 15 11 17  7  3  5 9
20: 1 13 17  9 11  3  7 19
...
T(7, 2) = 5 because A216319(7, 2) = 3 and Modd(3*5, 7) = 1 since floor(15/7) = 2 is even, hence Modd(3*5, 7) = mod(15, 7) = 1. The residue classes Modd 7 for  1, 3, 5 are shown in the array given in A113807 (including the negative numbers) [3]*[5] = [1] (Modd 7).
T(9, 2) = 7 because A216319(9, 2) = 5 and Modd(7*5, 9) = 1, since floor(35/9) = 3 is odd, hence Moddn(35, 9) = mod(-35, 9) = 1.

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012.

Cf. A055034, A203571, A187360, A216319.

rowa(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]); \\ A216319
Modd(x, n) = if ((x\n)%2, Mod(-x,n), Mod(x,n));
findinvm(k, n) = for (i=1, n, if (Modd(k*i, n) == 1, return(i)));
row(n) = my(ra=rowa(n)); vector(#ra, k, findinvm(ra[k], n)); \\ Michel Marcus, Sep 13 2023

T(n, k) = Inverse of A216319(n, k) (Modd n), for n >= 1. For Modd n see the comment above.

A216320 Irregular triangle: row n lists the Modd n order of the odd members of the reduced smallest nonnegative residue class modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 4, 4, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 5, 5, 5, 1, 2, 2, 2, 1, 3, 2, 6, 3, 6, 1, 3, 6, 3, 6, 2, 1, 4, 2, 4, 1, 8, 8, 4, 4, 8, 8, 2, 1, 8, 8, 8, 4, 8, 2, 4, 1, 6, 6, 3, 3, 2, 1, 9, 9, 3, 9, 3, 9, 9, 9, 1, 4, 4, 2, 2, 4, 4, 2, 1, 3, 6, 2, 3, 6

Offset: 1

Wolfdieter Lang, Sep 21 2012

The length of row n is delta(n):=A055034(n).
For the multiplicative group Modd n see a comment on A203571, and also on A216319.
A216319(n,k)^a(n,k) == +1 (Modd n), n >= 1.
If the Modd n order of an (odd) element from row n of A216319 is delta(n) (the row length) then this element is a primitive root Modd n. There is no primitive root Modd n if no such element of order delta(n) exists. For example, n = 12, 20, ... (see A206552 for more of these n values). There are phi(delta(n)) = A216321(n) such primitive roots Modd n if there exists one, where phi=A000010 (Euler's totient). The multiplicative group Modd n is cyclic if and only if there exists a primitive root Modd n. The multiplicative group Modd n is isomorphic to the Galois group G(Q(rho(n))/Q) with the algebraic number rho(n) := 2*cos(Pi/n), n>=1.

			The table a(n,k) begins:
  n\k 1  2  3  4  5  6  7  8  9 ...
  1   1
  2   1
  3   1
  4   1  2
  5   1  2
  6   1  2
  7   1  3  3
  8   1  4  4  2
  9   1  3  3
  10  1  4  4  2
  11  1  5  5  5  5
  12  1  2  2  2
  13  1  3  2  6  3  6
  14  1  3  6  3  6  2
  15  1  4  2  4
  16  1  8  8  4  4  8  8  2
  17  1  8  8  8  4  8  2  4
  18  1  6  6  3  3  2
  19  1  9  9  3  9  3  9  9  9
  20  1  4  4  2  2  4  4  2
  ...
a(7,2) = 3 because A216319(7,2) = 3 and 3^1 == 3 (Modd 7);
  3^2 = 9 == 5 (Modd 7) because floor(9/7)= 1 which is odd, therefore 9 (Modd 7) = -9 (mod 7) = 5; 3^3 == 5*3 (Modd n)
  = +1 because floor(15/7)=2 which is even, therefore 15 (Modd 7) = 15 (modd 7) = +1.
Row n=12 is the first row without an order = delta(n) (row length), in this case 4. Therefore there is no primitive root Modd 12, and the multiplicative group Modd 12 is non-cyclic.
  Its cycle structure is [[5,1],[7,1],[11,1]] which is the group Z_2 x Z_2 (the Klein 4-group).

Cf. A203571, A216321.

a(n,k) = order of A216319(n,k) Modd n, n>=1, k=1, 2, ..., A055034(n). This means: A216319(n,k)^a(n,k) == +1 (Modd n), n>=1, and a(n,k) is the smallest positive integer exponent satisfying this congruence. For Modd n see a comment on A203571.

A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

Original entry on oeis.org

0, 1, 4, 9, 13, 25, 36, 32, 64, 81, 66, 121, 124, 121, 196, 225, 170, 216, 324, 240, 400, 441, 272, 529, 513, 416, 676, 560, 522, 841, 900, 570, 792, 1089, 770, 1225, 1296, 752, 1170, 1521, 1093, 1681, 1376, 1232, 1936, 1656, 1410, 1728, 2304, 1490, 2500

Offset: 0

Wolfdieter Lang, May 01 2020

The smallest nonnegative reduced residue system modulo N is the ordered set RRS(N) (written as a list) with integers k from {0, 1, ..., N-1} satisfying gcd(k, N) = 1, for N >= 1. See A038566 (with A038566(1) = 0).
If only odd members of RRS(N) are considered, name this list RRSodd(N), e.g., RRSodd(1) = [], the empty list, RRSodd(2) = [1], etc. See A216319 (but there A216319(1) = 1). The number of elements of RRSodd(N) is delta(N) = A055034(N), for N >= 2, and 0 for N = 1.
Here only numbers N = 2*n + 1 >= 1 are considered, and for the empty list RRSodd(1) a(0) is set to 0.
a(n) gives for n >= 1 also the sum of the numbers of the primitive period of the unsigned Schick sequences SBB(2*n+1, q0 = 1) (BB for Brändli and Beyne), for which 2*n + 1 satisfies A135303(n) = 1 (in Schick's notation B(2*n+1) = 1, implying initial value q0 = 1). The numbers n satisfying A135303(n) = 1 are given in A333854.
The sequence with members gcd(a(n), 2*(2*n+1)) = A333849(n) is important for a length formula for the Euler tours ET(2*n+1, q0 = 1) given in A332441(n), for n >= 1 (but A333849(n) is used only for 2*n+1 values from A333854).

			n = 4: RRSodd(9) = {1, 5, 7} with sum a(4) = 13. Schick's unsigned cycle is SBB(9, 1) = (1, 7, 5). Because A135303(4) = B(9) = 1 there is only this cycle for n = 9.

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Cf. A038566, A055034, A135303, A216319, A333849, A333854.

{0}~Join~Table[Total@ Select[Range[1, m, 2], GCD[#, m] == 1 &], {m, Array[2 # + 1 &, 50]}] (* Michael De Vlieger, Oct 15 2020 *)

a(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m]))); \\ Michel Marcus, May 05 2020

apply( {A333848(n)=vecsum([2*m-1|m<-[1..n],gcd(m*2-1,n*2+1)==1])}, [0..50]) \\ M. F. Hasler, Jun 04 2020

a(n) = Sum_{j=1..delta(2*n+1)} RRSodd(2*n+1)_j, for n >= 1, with delta(k) = A055034(k). a(0) = 0 (undefined case).

A216321 phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 (degree of minimal polynomials with coefficients given in A187360).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 6, 4, 2, 4, 10, 4, 4, 4, 6, 4, 6, 4, 8, 8, 4, 8, 4, 4, 6, 6, 4, 8, 8, 4, 12, 8, 4, 10, 22, 8, 12, 8, 8, 8, 12, 6, 8, 8, 6, 12, 28, 8, 8, 8, 6, 16, 8, 8, 20, 16, 10, 8, 24, 8, 12, 12, 8, 12, 8, 8, 24, 16, 18, 16, 40, 8, 16, 12

Offset: 1

Wolfdieter Lang, Sep 21 2012

nonn

If n belongs to A206551 (cyclic multiplicative group Modd n) then there exist precisely a(n) primitive roots Modd n. For these n values the number of entries in row n of the table A216319 with value delta(n) (the row length) is a(n). Note that a(n) is also defined for the complementary n values from A206552 (non-cyclic multiplicative group Modd n) for which no primitive root Modd n exists.

See also A216322 for the number of primitive roots Modd n.

			a(8) = 2 because delta(8) = 4 and phi(4) = 2. There are 2 primitive roots Modd 8, namely 3 and 5 (see the two 4s in row n=8 of A216320). 8 = A206551(8).
a(12) = 2 because delta(12) = 4 and phi(4) = 2. But there is no primitive root Modd 12, because 4 does not show up in row n=12 of A216320. 12 = A206552(1).

Cf. A000010, A055034, A216319, A216320, A216322, A010554 (analog in modulo n case).

a(n)=eulerphi(ceil(eulerphi(2*n)/2)) \\ Charles R Greathouse IV, Feb 21 2013

a(n) = phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 with delta(1) = 1 and delta(n) = phi(2*n)/2 if n >= 2.

A126637 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x>y} of primitive Pythagorean triangles, sorted on values x+y (A126611), then on x-y.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, 1, 7, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 5, 11, 13, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 1, 5, 7, 11

Offset: 1

Lekraj Beedassy, Feb 08 2007

This sequence gives the consecutive rows n = 2*m + 1, for m >= 1, of the array A216319. See the example. - Wolfdieter Lang, Oct 24 2019

			From _Wolfdieter Lang_, Oct 24 2019: (Start)
From the array A216319 with n = 2*m + 1 = x + y, for m >= 1, the (x, y) values giving the terms of the present sequence as values x-y are:
m, n \ k    1      2      3      4      5      6 ...   x-y values
--------------------------------------------------------------------
1,  3:   (2,1)                                         1
2,  5:   (3,2) (4,1)                                   1 3
3,  7:   (4,3) (5,2)   (6,1)                           1 3  5
4,  9:   (5,4) (7,2)   (8,1)                           1 5  7
5, 11:   (6,5) (7,4)   (8,3)  (9,2) (10,1)             1 3  5  7  9
6, 13:   (7,6) (8,5)   (9,4) (10,3) (11,2) (12,1)      1 3  5  7  9  11
7, 15:   (8,7) (11,4) (13,2) (14,1)                    1 7 11 13
... (End)

Cf. A094192, A094193, A126611, A216319.

cp's OEIS Frontend

A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1.

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A216320 Irregular triangle: row n lists the Modd n order of the odd members of the reduced smallest nonnegative residue class modulo n.

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A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

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A216321 phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 (degree of minimal polynomials with coefficients given in A187360).

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A126637 Difference x-y of generator pairs (x,y) {x and y coprime and not both odd, x>y} of primitive Pythagorean triangles, sorted on values x+y (A126611), then on x-y.

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