login
Size of the key space for isomorphism verification of circulant graphs of order n.
5

%I #20 May 29 2017 12:57:09

%S 1,1,1,2,1,1,1,5,2,1,1,2,1,1,1,14,1,2,1,2,1,1,1,5,2,1,5,2,1,1,1,42,1,

%T 1,1,4,1,1,1,5,1,1,1,2,2,1,1,14,2,2,1,2,1,5,1,5,1,1,1,2,1,1,2,132,1,1,

%U 1,2,1,1,1,10,1,1,2,2,1,1,1,14,14,1,1,2,1,1,1,5,1,2

%N Size of the key space for isomorphism verification of circulant graphs of order n.

%C Multiplicative with a(p^m) = Catalan(m) (A000108). Coincides with A066060 up to n=63 except for n=32.

%D M. Muzychuk, A solution of the isomorphism problem for circulant graphs, Proc. London Math. Soc. (3) 88 (2004), no. 1, 1-41.

%H Antti Karttunen, <a href="/A069739/b069739.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = prod_p Catalan(m_p) where n=prod_p p^(m_p), p|n prime.

%F From _Antti Karttunen_, May 28-29 2017: (Start)

%F a(1) = 1; for n > 1, a(n) = A000108(A067029(n)) * a(A028234(n)).

%F a(n) = A246596(A156552(n)). (End)

%p A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:

%p A069739 := proc(n) local ifa; if n = 1 then 1; else ifa := ifactors(n)[2] ; mul( A000108(op(2,f)), f=ifa) ; end if; end proc:

%p seq(A069739(n),n=1..90) ; # _R. J. Mathar_, Feb 08 2011

%t Table[Times @@ Map[CatalanNumber, FactorInteger[n][[All, -1]]], {n, 90}] (* _Michael De Vlieger_, May 28 2017 *)

%o (Scheme) (define (A069739 n) (if (= 1 n) n (* (A000108 (A067029 n)) (A069739 (A028234 n))))) ;; _Antti Karttunen_, May 28 2017

%Y Cf. A000108, A066060, A028234, A067029, A156552, A246596.

%K mult,easy,nonn

%O 1,4

%A _Valery A. Liskovets_, Apr 15 2002