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A069739 Size of the key space for isomorphism verification of circulant graphs of order n.

%I A069739 #23 Nov 25 2024 09:06:53
%S A069739 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 A069739 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 A069739 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 A069739 Size of the key space for isomorphism verification of circulant graphs of order n.
%C A069739 Multiplicative with a(p^m) = Catalan(m) (A000108). Coincides with A066060 up to n=63 except for n=32.
%H A069739 Antti Karttunen, <a href="/A069739/b069739.txt">Table of n, a(n) for n = 1..10000</a>
%H A069739 M. Muzychuk, <a href="https://www.researchgate.net/publication/231890239_A_solution_of_the_isomorphism_problem_for_circulant_graphs">A solution of the isomorphism problem for circulant graphs</a>, Proc. London Math. Soc. (3) 88 (2004), no. 1, 1-41.
%H A069739 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A069739 a(n) = prod_p Catalan(m_p) where n=prod_p p^(m_p), p|n prime.
%F A069739 From _Antti Karttunen_, May 28-29 2017: (Start)
%F A069739 a(1) = 1; for n > 1, a(n) = A000108(A067029(n)) * a(A028234(n)).
%F A069739 a(n) = A246596(A156552(n)). (End)
%p A069739 A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A069739 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 A069739 seq(A069739(n),n=1..90) ; # _R. J. Mathar_, Feb 08 2011
%t A069739 Table[Times @@ Map[CatalanNumber, FactorInteger[n][[All, -1]]], {n, 90}] (* _Michael De Vlieger_, May 28 2017 *)
%o A069739 (Scheme) (define (A069739 n) (if (= 1 n) n (* (A000108 (A067029 n)) (A069739 (A028234 n))))) ;; _Antti Karttunen_, May 28 2017
%Y A069739 Cf. A000108, A066060, A028234, A067029, A156552, A246596.
%K A069739 mult,easy,nonn
%O A069739 1,4
%A A069739 _Valery A. Liskovets_, Apr 15 2002