A059998 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.

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, 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, 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

Offset: 1

Author

Naohiro Nomoto, Mar 10 2001

Keywords

• easy
• nonn
• nice

Comments

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,......

The largest possible value of a(n) is PrimePi(n)-1, which is frequently attained. - T. D. Noe, May 05 2008

Examples

			For n=9: 9 = 3+3+3 = 2+2+5, we can see 3 different primes. so a(9) = 3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..2000

Programs

Mathematica

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 *)

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

Formula

If n is of the form 2*(prime + 1) then a(n) is an even number.