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
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.
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).
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KD:= proc() local a; a:= (2*n)!/(n!*(n + 1)!)-1; if isprime(a) then return(a): fi; end: seq(KD(), n=1..150);
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Select[CatalanNumber[Range[200]]-1,PrimeQ] (* Harvey P. Dale, Dec 21 2019 *)
A240435
Catalan numbers whose sum of digits is a prime.
Original entry on oeis.org
2, 5, 14, 16796, 208012, 18367353072152, 14544636039226909, 3116285494907301262, 45950804324621742364, 176733862787006701400, 270557451039395118028642463289168566420671280440, 3935312233584004685417853572763349509774031680023800
Offset: 1
16796 is in sequence because Catalan(10) = (2*10)!/(10!*(10+1)!) = 16796: The digital sum(16796) = 29 which is prime.
208012 is in sequence because Catalan(12) = (2*12)!/(12!*(12+1)!) = 208012: The digital sum(208012) = 13 which is prime.
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KD := proc() local a, b; a:=(2*n)!/(n!*(n+1)!); b:=add( i,i = convert((a), base, 10))(a); if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..200);
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Select[CatalanNumber[Range[100]],PrimeQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Jul 19 2019 *)
A264042
Prime numbers adjacent to Catalan numbers (A000108).
Original entry on oeis.org
2, 3, 13, 41, 43, 131, 1429, 4861, 58787, 477638699, 4861946401451, 4861946401453, 5632681584560312734993915705849145099, 337485502510215975556783793455058624701, 4180080073556524734514695828170907458428751314321, 16435314834665426797069144960762886143367590394939
Offset: 1
A000108(5)=42 and both 41 and 43 are adjacent prime numbers, and thus are sequence members.
Cf.
A173868 (smallest primes > Catalan numbers).
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Select[Union[# - 1, # + 1] &@ Table[(2 n)!/n!/(n + 1)!, {n, 100}], PrimeQ] (* Michael De Vlieger, Nov 05 2015 *)
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A264042()={for (n=1,10000,if(n<3,q=binomial(2*n,n)/(n+1)); if(n>2,q=2*qold1*(8*qold2+qold1)/(10*qold2-qold1)); qold2=qold1; qold1=q; if(ispseudoprime(q-1),print1(q-1,"," )); if(ispseudoprime(q+1),print1(q+1,"," )));}
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C=1; for(n=1,1e3, C*=2*(2*n-1)/(n+1); if(ispseudoprime(C-1), print1(C-1", ")); if(ispseudoprime(C+1), print1(C+1", "))) \\ Charles R Greathouse IV, Nov 05 2015
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