

A294013


Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part prime.


2



0, 0, 2, 6, 10, 16, 22, 30, 38, 46, 54, 64, 74, 86, 98, 110, 122, 136, 150, 166, 182, 198, 214, 232, 250, 268, 286, 304, 322, 342, 362, 384, 406, 428, 450, 472, 494, 518, 542, 566, 590, 616, 642, 670, 698, 726, 754, 784, 814, 844, 874, 904, 934, 966, 998
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OFFSET

1,3


COMMENTS

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*xx^2 at prime values of x for x in 0 < x <= n. For example, d/dx 2*n*xx^2 = 2n2x. So for a(6), the prime values of x are x=2,3,5 and so a(6) = 122*2 + 122*3 + 122*5 = 8 + 6 + 2 = 16.  Wesley Ivan Hurt, Mar 24 2018


LINKS



FORMULA

a(n) = 2 * Sum_{i=1..n} (n  i) * A010051(i).


EXAMPLE

a(4) = 6: There are two partitions of 2*4 = 8 into two parts with the smaller part prime; they are (6,2) and (5,3). The sum of the differences of the parts is (6  2) + (5  3) = 4 + 2 = 6.


MATHEMATICA

Table[2 Sum[(n  i) (PrimePi[i]  PrimePi[i  1]), {i, n}], {n, 40}]
Table[Total[#[[1]]#[[2]]&/@Select[IntegerPartitions[2n, {2}], PrimeQ[ #[[2]]]&]], {n, 60}] (* Harvey P. Dale, Jan 25 2019 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



