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%I #13 Mar 14 2023 03:40:11
%S 0,0,0,0,0,1,2,2,3,3,4,4,3,3,5,4,6,5,5,5,7,5,8,6,7,7,9,6,8,5,8,7,10,5,
%T 11,8,10,9,10,4,12,7,11,9,13,7,14,8,13,11,15,9,14,7,14,11,16,7,15,8,
%U 15,13,17,6,18,11,17,13,17,5,19,11,18,13,20,10,21,11,20,15,20,9,22,10,21
%N Number of different primes occurring when n is expressed as p1+q1+r1 = ... = pk+qk+rk where pk,qk,rk are primes with pk <= qk <= rk.
%C Goldbach conjectured that every integer >5 is the sum of three primes. 6=2+2+2, 7=2+2+3, 8=2+3+3, 9=3+3+3=2+2+5,......
%C The largest possible value of a(n) is PrimePi(n)-1, which is frequently attained. - _T. D. Noe_, May 05 2008
%H T. D. Noe, <a href="/A059998/b059998.txt">Table of n, a(n) for n = 1..2000</a>
%F If n is of the form 2*(prime + 1) then a(n) is an even number.
%e For n=9: 9 = 3+3+3 = 2+2+5, we can see 3 different primes. so a(9) = 3.
%t a[n_] := Select[ Reverse /@ IntegerPartitions[n, {3}] , LessEqual @@ # && PrimeQ[#[[1]]] && PrimeQ[#[[2]]] && PrimeQ[#[[3]]] &] // Flatten // Union // Length; Table[a[n], {n, 1, 85}] (* _Jean-François Alcover_, Oct 03 2012 *)
%o (PARI) a(n)=my(v=List()); forprime(r=(n+2)\3,n-4, forprime(q=(n-r+1)\2,n-r-2, if(isprime(n-r-q), listput(v,r); listput(v,q); listput(v,n-r-q)))); #vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jul 14 2013
%K easy,nonn,nice
%O 1,7
%A _Naohiro Nomoto_, Mar 10 2001
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