

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
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OFFSET

1,2


COMMENTS

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*xx^2 such that 2nx is prime for integer values of x in 0 < x <= n. For example, d/dx 2*n*xx^2 = 2n2x. So for a(8), the integer values of x which make 16x prime are x=3,5 and so a(8) = 162*3 + 162*5 = 10 + 6 = 16.  Wesley Ivan Hurt, Mar 24 2018


LINKS

Table of n, a(n) for n=1..57.
Index entries for sequences related to partitions


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, (ni)*isprime(2*ni)); \\ 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



