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

%I A187129 #41 Apr 28 2023 21:36:58
%S A187129 2,3,5,12,7,18,24,24,30,47,49,55,40,59,48,100,102,50,89,120,109,136,
%T A187129 181,158,117,199,133,170,252,133,261,300,98,267,324,279,303,419,244,
%U A187129 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
%N A187129 Consider all pairs of primes (p,q) with p+q = 2n, p <= q; a(n) is the sum of all the q's.
%H A187129 Harvey P. Dale, <a href="/A187129/b187129.txt">Table of n, a(n) for n = 2..1000</a>
%F A187129 a(n) = Sum_{i=1..n} (2*n-i) * c(i) * c(2*n-i), where c = A010051. - _Wesley Ivan Hurt_, Apr 29 2021
%F A187129 a(n) = sopf(A362640(n)), n>=2. - _Wesley Ivan Hurt_, Apr 28 2023
%e A187129 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.
%p A187129 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
%t A187129 Table[Total[Select[IntegerPartitions[2*n,{2}],AllTrue[#,PrimeQ]&][[All,1]]],{n,2,100}] (* _Harvey P. Dale_, Aug 09 2020 *)
%o A187129 (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
%Y A187129 Cf. A045917, A185297, A186201, A362640.
%K A187129 nonn
%O A187129 2,1
%A A187129 _N. J. A. Sloane_, Mar 11 2011