The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A216320

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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.
(history; published version)

#16 by Alois P. Heinz at Tue Aug 11 15:41:47 EDT 2015

STATUS

proposed

approved

#15 by Bruno Berselli at Tue Aug 11 15:37:28 EDT 2015

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editing

proposed

#14 by Bruno Berselli at Tue Aug 11 15:37:22 EDT 2015

NAME

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

STATUS

approved

editing

#13 by Bruno Berselli at Mon Oct 01 18:18:42 EDT 2012

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proposed

approved

#12 by Wolfdieter Lang at Mon Oct 01 03:47:36 EDT 2012

STATUS

editing

proposed

#11 by Wolfdieter Lang at Mon Oct 01 03:47:26 EDT 2012

COMMENTS

If the Modd n order of an (odd) element from row n of A16319 A216319 is delta(n) (the row length) then this element is a primitive root of 1 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.

FORMULA

a(n,k) = order of A203571A216319(n,k) Modd n, n>=1, k=1, 2, ..., A055034(n). This means: A203571A216319(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.

EXAMPLE

a(7,2) = 3 because A203571A216319(7,2) = 3 and 3^1 == 3 (Modd 7);

Its cycle structure is in [[5,1],[7,1],[11,1]] which is the group Z_2 x Z_2 (the Klein 4-group).

STATUS

approved

editing

#10 by Alois P. Heinz at Thu Sep 27 17:18:30 EDT 2012

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proposed

approved

#9 by Wolfdieter Lang at Thu Sep 27 16:44:36 EDT 2012

STATUS

editing

proposed

#8 by Wolfdieter Lang at Thu Sep 27 16:44:26 EDT 2012

COMMENTS

For the multiplicative group Modd n (see a comment in on A203571), , and a comment also on A216319.

A216319(n,k)^a(n,k) = = +1 (Modd n), n >= 1.

STATUS

approved

editing

#7 by T. D. Noe at Sat Sep 22 17:33:16 EDT 2012

STATUS

editing

approved