A053429 -id:A053429 - cp's OEIS frontend

A230061 Primes of the form Catalan(n)+1.

2, 3, 43, 58787, 4861946401453, 337485502510215975556783793455058624701, 4180080073556524734514695828170907458428751314321, 1000134600800354781929399250536541864362461089950801, 944973797977428207852605870454939596837230758234904051

Offset: 1

K. D. Bajpai, Oct 08 2013

nonn

The 25th term a(25) in the sequence has 693 digits.

a(26) has 1335 digits; a(27) has 1647 digits; a(28) has 1694 digits; a(29) has 2554 digits; a(30) has 4857 digits; a(31) has 4876 digits; a(32) has 9641 digits. - Charles R Greathouse IV, Oct 09 2013

			a(3)= 43: Catalan(5)= (2*5)!/(5!*(5+1)!)= 42. Catalan(5)+1= 43 which is prime.
a(4)= 58787: Catalan(11)= (2*11)!/(11!*(11+1)!)= 58786. Catalan(11)+1= 58787 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..25
Chris K. Caldwell and G. L. Honaker,Jr., Prime Curios! 4250

Cf. A000108, A024492, A141351.

Cf. A053429 (numbers n such that Catalan(n)+1 is prime).

KD:= proc() local a,b,c; a:= (2*n)!/(n!*(n + 1)!); b:=a+1;if isprime(b) then return(b): fi; end: seq(KD(),n=1..50);

Select[CatalanNumber[Range[100]]+1,PrimeQ] (* Harvey P. Dale, Aug 26 2021 *)

for(n=1,1e3,if(ispseudoprime(t=binomial(2*n,n)/(n+1)+1),print1(t", "))) \\ Charles R Greathouse IV, Oct 08 2013

A231885 Primes of the form Catalan(n) - 1.

Original entry on oeis.org

13, 41, 131, 1429, 4861, 477638699, 4861946401451, 5632681584560312734993915705849145099, 16435314834665426797069144960762886143367590394939, 171069509209912116706646841207804116182333282333996796075729541331934805254423

Offset: 1

K. D. Bajpai, Nov 21 2013

nonn

The 22nd term a(22) in the sequence has 862 digits.

a(23) has 1134 digits; a(25) has 1413 digits; a(30) has 2046 digits; a(31) has 2348 digits (these are not included in b-file).

			a(2)= 41: n= 5: (2*n)!/(n!*(n + 1)!)-1= 41 which is prime.
a(4)= 1429: n= 8: (2*n)!/(n!*(n + 1)!)-1= 1429 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..22

Cf.  A000108 (Catalan numbers).
Cf.  A053427 (numbers n : Catalan(n)-1 is prime).
Cf.  A053429 (numbers n such that Catalan(n)+1 is prime).
Cf.  A230061 (primes of the form Catalan(n)+1).

KD:= proc() local a; a:= (2*n)!/(n!*(n + 1)!)-1;  if isprime(a) then return(a):  fi;  end:  seq(KD(), n=1..150);

Select[CatalanNumber[Range[200]]-1,PrimeQ] (* Harvey P. Dale, Dec 21 2019 *)

A053427 Numbers n such that Catalan(n)-1 is prime.

Original entry on oeis.org

4, 5, 6, 8, 9, 18, 25, 66, 87, 134, 145, 200, 384, 443, 502, 589, 625, 638, 1082, 1235, 1236, 1439, 1892, 2014, 2355, 2380, 2592, 2676, 2981, 3406, 3908, 4775, 5885, 10617, 16108, 17035, 18164, 18307, 20565, 24542, 26388, 32786, 47379, 49711, 50103, 55067

Offset: 1

David Broadhurst, Jan 10 2000

nonn

Primality up to Catalan(5885)-1 proved by PrimeForm.

The next term, if it exists, is > 60000. - Vaclav Kotesovec, Apr 26 2021

			Catalan(25)-1 = 50!/25!/26!-1 = 4861946401451 is prime.

Cf. A000108, A053429.

Reap[ Do[ If[ PrimeQ[ CatalanNumber[n] - 1], Print[n]; Sow[n]], {n, 0, 10^4}]][[2, 1]] (* Jean-François Alcover, Feb 02 2015 *)

is(n)=ispseudoprime(binomial(2*n,n)/(n+1)-1) \\ Charles R Greathouse IV, Jan 03 2014

ABC2 C(2*$a,$a)/($a+1)-1
a: from 5886 to 100000
Charles R Greathouse IV, Jan 03 2014

a(34)-a(41) from Charles R Greathouse IV, Jan 03 2014
a(42) from Vaclav Kotesovec, Apr 20 2021
a(43)-a(46) from Vaclav Kotesovec, Apr 25 2021

A235051 a(n) = |{0 < k < n-2: C(sigma(k) + phi(n-k)/2) - 1 is prime}|, where C(j) is the j-th Catalan number (A000108), sigma(k) is the sum of all positive divisors of k, and phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 3, 5, 3, 4, 4, 4, 5, 3, 4, 4, 5, 3, 2, 1, 4, 2, 2, 2, 4, 2, 2, 2, 2, 4, 5, 1, 1, 1, 3, 1, 2, 2, 5, 1, 1, 3, 1, 2, 1, 2, 1, 4, 3, 3, 3, 1, 0, 0, 2, 2, 2, 1, 7, 1, 0, 4, 1, 3, 1, 1, 2, 2, 1, 7, 4, 4, 1, 3, 3, 2, 3, 4, 3, 1, 7, 1, 5, 2, 5, 1, 3, 3, 4, 5, 1, 4, 2, 3, 4, 6, 5, 3

Offset: 1

Zhi-Wei Sun, Jan 02 2014

nonn

It might seem that a(n) > 0 for all n > 63, but 9122 and 9438 are counterexamples.

			a(22) = 1 since sigma(8) + phi(14)/2 = 15 + 6/2 = 18 with C(18) - 1 = 477638699 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1800

Cf. A000010, A000040, A000108, A000203, A053427, A053429, A231885, A234963.

sigma[n_]:=DivisorSigma[1,n]
f[n_,k_]:=CatalanNumber[sigma[k]+EulerPhi[n-k]/2]-1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

C(n)=binomial(2*n, n)/(n+1)
a(n)=sum(k=1,n-3,ispseudoprime(C(sigma(k)+eulerphi(n-k)/2)-1)) \\ Charles R Greathouse IV, Jan 03 2014

cp's OEIS Frontend

A230061 Primes of the form Catalan(n)+1.

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A231885 Primes of the form Catalan(n) - 1.

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A053427 Numbers n such that Catalan(n)-1 is prime.

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A235051 a(n) = |{0 < k < n-2: C(sigma(k) + phi(n-k)/2) - 1 is prime}|, where C(j) is the j-th Catalan number (A000108), sigma(k) is the sum of all positive divisors of k, and phi(.) is Euler's totient function.

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