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

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, 34, 35, 37, 16, 41, 38, 30, 39, 43, 46, 47, 25, 49, 44, 27, 24, 36, 40, 42, 45, 51, 50, 52, 33, 53, 54, 55, 48, 57, 58, 59, 60, 61, 56, 63, 32, 65, 66, 67, 68, 69

Offset: 1

Author

N. J. A. Sloane, Jul 25 2019

Keywords

• nonn

Comments

Conjecture: This is a permutation of the positive integers. [The conjecture is true, see A309364. - Rémy Sigrist, Jul 25 2019]

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.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A309200

Crossrefs

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

Programs

Maple

with(numtheory);
# the general transformation
f := proc(b) local t1,d,j,dlis,L,hit,i,n,a,n1;
if whattype(b) <> list then RETURN([]); fi;
n1:=nops(b); a:=[]; L:=10000;
hit:=Array(0..L,0);
for n from 1 to n1 do
   t1:=b[n];
dlis:=sort(convert(divisors(t1),list));
for j from 1 to nops(dlis) do d:=dlis[j];
  if d > L then error("d too large",n,t1,d); fi;
   if hit[d]=0 then break; fi; od:
a:=[op(a),d];
hit[d]:=1;          od;
[seq(a[i],i=1..nops(a))];
end;
# the Catalan numbers
C:=[seq(binomial(2*n,n)/(n+1),n=1..40)];
f(C);

PARI

\\ See Links section.

Sage

def transform(sup, fun):
    A = []
    for n in (1..sup):
        D = divisors(fun(n))
        A.append(next(d for d in D if d not in A))
    return A
A309200list = lambda lim: transform(lim, catalan_number)
print(A309200list(29)) # Peter Luschny, Jul 26 2019

Extensions

More terms from Rémy Sigrist, Jul 25 2019