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a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence.
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%I #22 Apr 28 2020 15:10:00

%S 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 34,35,37,16,41,38,30,39,43,46,47,25,49,44,27,24,36,40,42,45,51,50,52,

%U 33,53,54,55,48,57,58,59,60,61,56,63,32,65,66,67,68,69

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

%C Conjecture: This is a permutation of the positive integers. [The conjecture is true, see A309364. - _Rémy Sigrist_, Jul 25 2019]

%C 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 Rémy Sigrist, <a href="/A309200/b309200.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A309200/a309200.gp.txt">PARI program for A309200</a>

%p with(numtheory);

%p # the general transformation

%p f := proc(b) local t1,d,j,dlis,L,hit,i,n,a,n1;

%p if whattype(b) <> list then RETURN([]); fi;

%p n1:=nops(b); a:=[]; L:=10000;

%p hit:=Array(0..L,0);

%p for n from 1 to n1 do

%p t1:=b[n];

%p dlis:=sort(convert(divisors(t1),list));

%p for j from 1 to nops(dlis) do d:=dlis[j];

%p if d > L then error("d too large",n,t1,d); fi;

%p if hit[d]=0 then break; fi; od:

%p a:=[op(a),d];

%p hit[d]:=1; od;

%p [seq(a[i],i=1..nops(a))];

%p end;

%p # the Catalan numbers

%p C:=[seq(binomial(2*n,n)/(n+1),n=1..40)];

%p f(C);

%o (PARI) \\ See Links section.

%o (Sage)

%o def transform(sup, fun):

%o A = []

%o for n in (1..sup):

%o D = divisors(fun(n))

%o A.append(next(d for d in D if d not in A))

%o return A

%o A309200list = lambda lim: transform(lim, catalan_number)

%o print(A309200list(29)) # _Peter Luschny_, Jul 26 2019

%Y Cf. A000027, A000108, A000217, A111273, A309364..

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jul 25 2019

%E More terms from _Rémy Sigrist_, Jul 25 2019