A023816 -id:A023816 - cp's OEIS frontend

A023847 Sum of exponents in prime-power factorization of binomial(5n, 2n).

0, 2, 4, 4, 6, 8, 8, 11, 11, 11, 13, 13, 13, 18, 16, 17, 17, 19, 18, 18, 22, 24, 21, 23, 23, 24, 28, 26, 26, 30, 29, 32, 28, 30, 31, 31, 32, 35, 35, 36, 36, 36, 37, 33, 35, 38, 36, 39, 36, 40, 40, 41, 45, 48, 43, 46, 46, 45, 50, 47, 49, 52, 52, 49, 46, 51, 51, 50, 50, 55, 51, 57, 54, 57, 57, 55, 59, 62

Offset: 0

Clark Kimberling

Ivan Neretin, Table of n, a(n) for n = 0..10000 (corrected by Sean A. Irvine, Jan 18 2019)

Cf. A001222, A001450, A022559.

Cf. A023816-A023854.

Table[PrimeOmega[Binomial[5 n, 2 n]], {n, 77}] (* Ivan Neretin, Nov 09 2017 *)

a(n) = bigomega(binomial(5*n, 2*n)); \\ Michel Marcus, Nov 09 2017

a(n) = my(res = 0); forprime(p = 2, 5*n, res += (val(5*n, p) - val(2*n, p) - val(3*n, p))); res
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Nov 09 2017

a(n) = A001222(A001450(n)). - Michel Marcus, Nov 09 2017

a(0) = 0 prepended by David A. Corneth, Nov 09 2017

A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A001222, A022559, A023816, A023834.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Binomial[4*n, 2*n]] - PrimeOmega[Binomial[2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = my(v = binomial(4*n, 2*n)/binomial(2*n, n)); bigomega(numerator(v)) - bigomega(denominator(v)); \\ Michel Marcus, Sep 30 2013

vp(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(s);forprime(p=2,4*n,s+=vp(4*n,p)-3*vp(2*n,p)+2*vp(n,p)); s \\ Charles R Greathouse IV, Sep 30 2013

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023834(n) - A023816(n).
a(n) = A022559(4*n) - 3*A022559(2*n) + 2*A022559(n). (End)

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023834 Sum of exponents in prime-power factorization of C(4n,2n).

Original entry on oeis.org

0, 2, 3, 5, 6, 6, 7, 11, 9, 11, 12, 12, 13, 15, 14, 15, 15, 16, 15, 18, 17, 20, 22, 22, 22, 23, 22, 23, 22, 23, 25, 28, 25, 25, 29, 28, 28, 32, 30, 32, 31, 30, 31, 34, 34, 34, 34, 36, 35, 37, 34, 36, 38, 38, 38, 40, 38, 41, 42, 42, 40, 44, 46, 44, 43, 43, 44, 46, 42, 46, 47, 47, 46, 48, 47, 50, 51, 50

Offset: 0

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A001222, A001448, A023816, A022559.

Join[{0}, Table[Total[FactorInteger[Binomial[4 n, 2 n]][[All, 2]]], {n, 77}]] (* Ivan Neretin, Oct 26 2017 *)

a(n) = bigomega(binomial(4*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001448(n)).
a(n) = A023816(2*n).
a(n) = A022559(4*n) - 2*A022559(2*n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 7, 9, 7, 8, 8, 10, 9, 10, 10, 11, 11, 11, 12, 12, 11, 13, 13, 14, 11, 13, 14, 14, 13, 14, 14, 16, 15, 16, 18, 19, 19, 19, 19, 21, 19, 20, 19, 21, 20, 21, 21, 21, 19, 20, 20, 22, 22, 24, 25, 25, 23, 23, 23, 24, 24, 27, 26, 27, 25, 27, 28, 29, 28

Offset: 0

Labos Elemer, Mar 28 2003

It is easy to show that a(n) is between n/log(n) and 2n/log(n) (for n>n0), cf. [Campbell 1984]. The sequence A137687, roughly the middle of this interval, is a fair 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. A001222, A000108, A022559, A023816, A048621, A081405, A120626, A137686-A137687.

Cf. A080405.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(2*n,n)/(1+n)) fi end: seq(a(n), n=0..75); # Zerinvary Lajos, Apr 11 2008

a[n_] := PrimeOmega[ CatalanNumber[n]]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Jul 02 2013 *)

A081399(n)=bigomega(prod(i=2,n,(n+i)/i)) \\ M. F. Hasler, Feb 06 2008

a(n)=A001222[A000108(n)]

Edited and extended by M. F. Hasler, Feb 06 2008

A023817 Sum of exponents in prime-power factorization of C(2n,n-1).

Original entry on oeis.org

0, 2, 2, 4, 4, 6, 4, 7, 6, 7, 6, 9, 8, 11, 9, 12, 9, 13, 10, 13, 12, 13, 12, 15, 14, 14, 14, 16, 14, 17, 12, 18, 16, 16, 15, 18, 15, 18, 17, 20, 19, 22, 20, 22, 22, 23, 20, 25, 21, 24, 22, 24, 22, 25, 21, 24, 22, 24, 23, 28, 26, 27, 26, 29, 25, 27, 25, 30, 28, 30, 26, 32, 29, 31, 31, 31, 31, 34, 29, 32

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A001791, A023816, A023818, A076191.

Join[{0},Total[FactorInteger[#][[All,2]]]&/@Table[Binomial[2n,n-1], {n,2,80}]] (* Harvey P. Dale, Sep 14 2016 *)
Table[PrimeOmega[Binomial[2 n, n + 1]], {n, 1, 200}] (* Clark Kimberling, May 04 2025 *)

a(n) = bigomega(binomial(2*n, n-1)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001791(n)).
a(n) = A023816(n) - A076191(n). (End)

A023818 Sum of exponents in prime-power factorization of C(2n,n-2).

Original entry on oeis.org

0, 2, 3, 5, 4, 4, 6, 8, 6, 7, 8, 9, 8, 10, 11, 12, 11, 11, 12, 14, 11, 12, 14, 15, 13, 15, 16, 16, 13, 13, 17, 19, 14, 16, 18, 17, 15, 18, 19, 22, 20, 20, 21, 24, 21, 20, 23, 24, 23, 24, 22, 23, 22, 23, 24, 25, 22, 24, 27, 27, 22, 26, 29, 30, 26, 26, 28, 30, 27, 28, 31, 31, 29, 31, 31, 33, 31, 28, 31

Offset: 2

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A001222, A002694, A023816, A023817.

Join[{0}, Table[Total[FactorInteger[Binomial[2 n, n - 2]][[All, 2]]], {n, 3, 80}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[2*n, n-2]]; Array[a, 100, 2] (* Amiram Eldar, Jun 12 2025 *)

a(n) = bigomega(binomial(2*n, n-2)); \\ Amiram Eldar, Jun 12 2025

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A002694(n)).
a(n) = A023817(n) - A022559(n+2) + A022559(n-1). (End)

Offset corrected to 2 by Ivan Neretin, Nov 02 2017

cp's OEIS Frontend

A023847 Sum of exponents in prime-power factorization of binomial(5n, 2n).

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A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

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A023834 Sum of exponents in prime-power factorization of C(4n,2n).

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A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

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A023817 Sum of exponents in prime-power factorization of C(2n,n-1).

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A023818 Sum of exponents in prime-power factorization of C(2n,n-2).

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