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A309200 a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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);

PROG

(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

CROSSREFS

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

Sequence in context: A291756 A286463 A286362 * A097964 A133133 A024710

Adjacent sequences:  A309197 A309198 A309199 * A309201 A309202 A309203

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 25 2019

EXTENSIONS

More terms from Rémy Sigrist, Jul 25 2019

STATUS

approved

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Last modified May 18 22:57 EDT 2021. Contains 344007 sequences. (Running on oeis4.)