A081399 -id:A081399 - cp's OEIS frontend

A137686 a(n) = Bigomega(Catalan(n)) - round( 3 n /(2 log(n+2))) (= A081399 - A137687).

0, -1, -1, -2, -1, -1, 0, -2, -1, -2, -1, -1, -1, 0, 1, -1, 0, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, -1, 1, 0, 1, -2, -1, 0, 0, -2, -1, -1, 1, -1, 0, 2, 2, 2, 2, 1, 3, 1, 2, 0, 2, 1, 1, 1, 1, -1, -1, -1, 1, 0, 2, 3, 3, 0, 0, 0, 1, 0, 3, 2, 2, 0, 2, 3, 3, 2, 2, 3, 4, 1, 0, 1, 1, 1, 1, 1, 3, 1, 4, 2, 2, 1, 2, 2, 3, 2, 3, 1, 2, 0, 1, 0, 2, 1, 2, 2, 3, 1, 3, 2, 3, 1, 2, 3, 3, 2, 3

Offset: 0

M. F. Hasler, Feb 06 2008

It is easy to show that A081399(n) = bigomega(Catalan(n)) is between n/log(n) and 2n/log(n) (for n>n0). The sequence A137687 is roughly the middle of this interval, which turns out to be a fair approximation to A081399. The present sequence lists the (signed) difference.

M. F. Hasler, Table of n, a(n) for n = 0..3000.
Douglas M. Campbell, The Computation of Catalan Numbers, Mathematics Magazine, Vol. 57, No. 4. (Sep., 1984), pp. 195-208.

Cf. A000108, A081399, A120626, A137687.

A137686(n) = bigomega(prod(i=2,n,(n+i)/i)) - round(3*n/log(n+2)/2)

a(n) = A001222(A000108(n)).

A137687 a(n) = round(3 n / (2 log(n+2))), an approximation to A081399.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26

Offset: 0

M. F. Hasler, Feb 06 2008

It is easy to show that A081399(n) is between n/log(n) and 2n/log(n) (for n>n0), cf. [Campbell 1984]. This sequence A137687 is roughly the middle of this interval (with log(n) replaced by log(n+2) to be well-defined for all n>=0), which turns out to be a fair (and simple, increasing) approximation for A081399.

See A137686 for the (signed) difference of the two sequences.

M. F. Hasler, Table of n, a(n) for n = 0..3000.
Douglas M. Campbell, The Computation of Catalan Numbers, Mathematics Magazine, Vol. 57, No. 4. (Sep., 1984), pp. 195-208.

Cf. A000108, A001222, A081399, A120626, A137686.

A137687(n) = round(3*n/log(n+2)/2) \\ M. F. Hasler, Feb 06 2008

A080405 Number of distinct primes dividing n-th Catalan number.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 17, 17, 19, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 22, 22, 23, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25

Offset: 0

Labos Elemer, Mar 19 2003

nonn

			a(5) = 3, as C(5) = 42 factors as 2*3*7 (3 distinct prime factors).

Enrique Pérez Herrero, Table of n, a(n) for n = 0..3000

Cf. A001221, A000108.

Cf. A081399.

PrimeNu[CatalanNumber[Range[0,80]]] (* Harvey P. Dale, Mar 27 2013 *)

C(n)=binomial(2*n,n)/(n+1);
for(n=1,100,print1(matsize(factor(C(n-1)))[1],", ")); \\ Joerg Arndt, Apr 19 2014

a(n) = A001221(A000108(n)).

A121612 Numbers k such that the number of prime divisors of the k-th Catalan number (counted with multiplicity) divides k.

Original entry on oeis.org

2, 3, 4, 8, 10, 12, 16, 20, 22, 117, 408, 432, 444, 492, 504, 508, 1555, 1560, 1605, 1675, 1790, 5832, 5940, 5976, 6048, 6078, 6102, 6108, 6132, 6138, 21175, 21266, 21280, 21301, 21434, 21462, 21469, 21616, 21623, 21749, 21770, 21784, 81472, 81528, 81648

Offset: 1

Jonathan Vos Post, Sep 08 2006

nonn

a(46) > 200000, if it exists. - Amiram Eldar, Oct 10 2024

			a(1) = 2 because bigomega(Catalan(2)) = bigomega(2) = 1 and 1 | 2.

Cf. A000108, A001222, A081399.

fQ[n_] := Mod[n, Plus @@ (Last@# & /@ FactorInteger[ Binomial[2n, n]/(n + 1)])] == 0; s = {}; Do[ If[fQ@n, Print@n; AppendTo[s, n]], {n, 2, 35500}] (* Robert G. Wilson v, Sep 11 2006 *)

{k such that A081399(k)|k} = {k such that A001222(A000108(k))|k}.

Corrected and extended (a(10)-a(42)) by Robert G. Wilson v, Sep 11 2006

a(43)-a(45) from Amiram Eldar, Oct 10 2024

cp's OEIS Frontend

A137686 a(n) = Bigomega(Catalan(n)) - round( 3 n /(2 log(n+2))) (= A081399 - A137687).

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A137687 a(n) = round(3 n / (2 log(n+2))), an approximation to A081399.

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PARI

A080405 Number of distinct primes dividing n-th Catalan number.

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PARI

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A121612 Numbers k such that the number of prime divisors of the k-th Catalan number (counted with multiplicity) divides k.

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