A187129 Consider all pairs of primes (p,q) with p+q = 2n, p <= q; a(n) is the sum of all the q's.

2, 3, 5, 12, 7, 18, 24, 24, 30, 47, 49, 55, 40, 59, 48, 100, 102, 50, 89, 120, 109, 136, 181, 158, 117, 199, 133, 170, 252, 133, 261, 300, 98, 267, 324, 279, 303, 419, 244, 303, 494, 345, 260, 593, 302, 343, 503, 207, 452, 612, 399, 488, 668, 526, 619, 872, 574, 540, 1082, 352, 475, 920, 273, 691, 865, 598, 523, 822, 725, 864, 1211

Offset: 2

Author

N. J. A. Sloane, Mar 11 2011

Keywords

• nonn

Examples

			2*5 = 10 can be expressed as the sum of two primes in two ways: 3+7 and 5+5, so a(5) = 7+5 = 12.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A045917, A185297, A186201, A362640.

Programs

Maple

with(numtheory); a:=n-> sum( (2*n-i)*( ((pi(i) - pi(i-1)) * (pi(2*n-i) - pi(2*n-i-1))) ), i = 1..n ); seq(a(k),k=1..100); # Wesley Ivan Hurt, Jan 20 2013

Mathematica

Table[Total[Select[IntegerPartitions[2*n,{2}],AllTrue[#,PrimeQ]&][[All,1]]],{n,2,100}] (* Harvey P. Dale, Aug 09 2020 *)

PARI

a(n) = my(s=0); forprime(p=1, n, if (isprime(2*n-p), s += 2*n-p)); s; \\ Michel Marcus, Apr 29 2021

Formula

a(n) = Sum_{i=1..n} (2*n-i) * c(i) * c(2*n-i), where c = A010051. - Wesley Ivan Hurt, Apr 29 2021

a(n) = sopf(A362640(n)), n>=2. - Wesley Ivan Hurt, Apr 28 2023