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A309200 a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence.

%I A309200 #22 Apr 28 2020 15:10:00
%S A309200 1,2,5,7,3,4,11,10,13,17,14,19,20,6,9,15,22,12,21,23,26,8,18,29,28,31,
%T A309200 34,35,37,16,41,38,30,39,43,46,47,25,49,44,27,24,36,40,42,45,51,50,52,
%U A309200 33,53,54,55,48,57,58,59,60,61,56,63,32,65,66,67,68,69
%N A309200 a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence.
%C A309200 Conjecture: This is a permutation of the positive integers. [The conjecture is true, see A309364. - _Rémy Sigrist_, Jul 25 2019]
%C A309200 Given any monotonically increasing sequence {b(n): n >= 1} of positive integers we can define a sequence {a(n): n >= 1} by setting a(n) to be smallest divisor of b(n) not already in the {a(n)} sequence. The triangular numbers A000217 produce A111273. A000027 is fixed under this transformation.
%H A309200 Rémy Sigrist, <a href="/A309200/b309200.txt">Table of n, a(n) for n = 1..10000</a>
%H A309200 Rémy Sigrist, <a href="/A309200/a309200.gp.txt">PARI program for A309200</a>
%p A309200 with(numtheory);
%p A309200 # the general transformation
%p A309200 f := proc(b) local t1,d,j,dlis,L,hit,i,n,a,n1;
%p A309200 if whattype(b) <> list then RETURN([]); fi;
%p A309200 n1:=nops(b); a:=[]; L:=10000;
%p A309200 hit:=Array(0..L,0);
%p A309200 for n from 1 to n1 do
%p A309200    t1:=b[n];
%p A309200 dlis:=sort(convert(divisors(t1),list));
%p A309200 for j from 1 to nops(dlis) do d:=dlis[j];
%p A309200   if d > L then error("d too large",n,t1,d); fi;
%p A309200    if hit[d]=0 then break; fi; od:
%p A309200 a:=[op(a),d];
%p A309200 hit[d]:=1;          od;
%p A309200 [seq(a[i],i=1..nops(a))];
%p A309200 end;
%p A309200 # the Catalan numbers
%p A309200 C:=[seq(binomial(2*n,n)/(n+1),n=1..40)];
%p A309200 f(C);
%o A309200 (PARI) \\ See Links section.
%o A309200 (Sage)
%o A309200 def transform(sup, fun):
%o A309200     A = []
%o A309200     for n in (1..sup):
%o A309200         D = divisors(fun(n))
%o A309200         A.append(next(d for d in D if d not in A))
%o A309200     return A
%o A309200 A309200list = lambda lim: transform(lim, catalan_number)
%o A309200 print(A309200list(29)) # _Peter Luschny_, Jul 26 2019
%Y A309200 Cf. A000027, A000108, A000217, A111273, A309364..
%K A309200 nonn
%O A309200 1,2
%A A309200 _N. J. A. Sloane_, Jul 25 2019
%E A309200 More terms from _Rémy Sigrist_, Jul 25 2019