A309200 - OEIS

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

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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: A361118 A361130 A361128 * A361170 A097964 A133133

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