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%I #62 Apr 16 2018 18:59:00
%S 0,2,4,8,4,12,20,16,28,40,32,48,40,34,56,78,68,60,88,80,112,144,132,
%T 168,156,144,184,170,156,202,248,234,220,272,256,310,364,346,328,388,
%U 368,432,412,394,464,444,424,406,484,464,544,624,600,684,768,742,828
%N Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part prime.
%C Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 such that 2n-x is prime for integer values of x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(8), the integer values of x which make 16-x prime are x=3,5 and so a(8) = 16-2*3 + 16-2*5 = 10 + 6 = 16. - _Wesley Ivan Hurt_, Mar 24 2018
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = 2 * Sum_{i=1..n} (n - i)*A010051(2n - i).
%e a(4) = 8; there are two partitions of 2*4 = 8 into two parts with the larger part prime: (7,1) and (5,3). The sum of the differences of the parts is (7 - 1) + (5 - 3) = 6 + 2 = 8.
%t Table[2 Sum[(n - i) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 60}]
%o (PARI) a(n) = 2*sum(i=1, n, (n-i)*isprime(2*n-i)); \\ _Michel Marcus_, Mar 25 2018
%Y Cf. A010051, A294013.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Oct 21 2017
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