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 A290288 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part prime. 1
 0, 2, 4, 8, 4, 12, 20, 16, 28, 40, 32, 48, 40, 34, 56, 78, 68, 60, 88, 80, 112, 144, 132, 168, 156, 144, 184, 170, 156, 202, 248, 234, 220, 272, 256, 310, 364, 346, 328, 388, 368, 432, 412, 394, 464, 444, 424, 406, 484, 464, 544, 624, 600, 684, 768, 742, 828 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS FORMULA a(n) = 2 * Sum_{i=1..n} (n - i)*A010051(2n - i). EXAMPLE 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. MATHEMATICA Table[2 Sum[(n - i) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 60}] PROG (PARI) a(n) = 2*sum(i=1, n, (n-i)*isprime(2*n-i)); \\ Michel Marcus, Mar 25 2018 CROSSREFS Cf. A010051, A294013. Sequence in context: A016635 A133992 A294094 * A126215 A165617 A273170 Adjacent sequences:  A290285 A290286 A290287 * A290289 A290290 A290291 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Oct 21 2017 STATUS approved

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Last modified April 18 10:21 EDT 2021. Contains 343087 sequences. (Running on oeis4.)