A327395 - OEIS

%I #5 Sep 16 2019 12:38:28

%S 1,1,1,2,1,2,1,4,1,2,1,4,1,2,3,8,1,2,1,4,1,2,1,8,1,2,1,4,1,6,1,16,3,2,

%T 5,4,1,2,1,8,1,2,1,4,5,2,1,16,1,2,3,4,1,2,5,8,1,2,1,12,1,2,1,32,1,6,1,

%U 4,3,10,1,8,1,2,3,4,7,2,1,16,1,2,1,4,5

%N Quotient of n over the maximum connected divisor of n.

%C Requires A305079(n) steps to reach 1, the only fixed point.

%C A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%F a(n) = n/A327076(n).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t maxcon[n_]:=Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]];

%t Table[n/maxcon[n],{n,100}]

%Y See link for additional crossrefs.

%Y Positions of 1's are A305078.

%Y Positions of 2's are 2 * A305078.

%Y Cf. A000005, A304716, A304714, A305079, A327076.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 15 2019