

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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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
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OFFSET

1,5


COMMENTS

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).


LINKS



FORMULA

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


EXAMPLE

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.


PROG

(PARI) 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


CROSSREFS



KEYWORD

nonn,tabf,easy


AUTHOR



STATUS

approved



