A268341 Triangle T(n,k) = Degree of vertex k in the unitary addition Cayley graph Gn, 0<=k<=n-1, with T(1,0)=0.

0, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 6, 5, 5, 6, 5, 5, 6, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset

1,4

Comments

For n>1, the unitary addition Cayley graph Gn is the graph whose vertices are Z/nZ and where 2 vertices x and y are adjacent if x+y is a unit in Z/nZ.

Links

Table of n, a(n) for n=1..91.

M. Deaconescu, Adding units mod n, Elem. Math. 55 (2000) 123-127.

J. W. Sander, On the addition of units and nonunits mod m, Journal of Number Theory, Volume 129, Issue 10, October 2009, Pages 2260-2266.

Deepa Sinha, Pravin Garg and Anjali Singh, Some properties of unitary addition Cayley graphs, Notes on Number Theory and Discrete Mathematics, Volume 17, 2011, Number 3, Pages 49—59. See Figure 1 p. 3.

Formula

T(n,k) = phi(n) if n is even or if n id odd and gcd(n,k) != 1, phi(n-1) if n is odd and gcd(n,k) = 1, where phi is the Euler totient function.

Example

Array starts:
0;
1, 1;
2, 1, 1;
2, 2, 2, 2;
4, 3, 3, 3, 3;
2, 2, 2, 2, 2, 2;
6, 5, 5, 5, 5, 5, 5;
...

Mathematica

Table[Which[EvenQ@ n, EulerPhi@ n, OddQ@ n && ! CoprimeQ[n, k], EulerPhi@ n, OddQ@ n && CoprimeQ[n, k], EulerPhi[n] - 1], {n, 13}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Feb 02 2016 *)

Prog

(PARI) T(n, k) = if (n % 2, if (gcd(n, k)==1, eulerphi(n)-1, eulerphi(n)), eulerphi(n));

Crossrefs

Sequence in context: A029285 A134337 A261733 * A053633 A216460 A156755

Adjacent sequences: A268338 A268339 A268340 * A268342 A268343 A268344

Keyword

nonn,tabl

Author

Michel Marcus, Feb 02 2016

Status

approved