A294013 - OEIS

%I #22 Jan 25 2019 17:35:12

%S 0,0,2,6,10,16,22,30,38,46,54,64,74,86,98,110,122,136,150,166,182,198,

%T 214,232,250,268,286,304,322,342,362,384,406,428,450,472,494,518,542,

%U 566,590,616,642,670,698,726,754,784,814,844,874,904,934,966,998

%N Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller 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 at prime values of x for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(6), the prime values of x are x=2,3,5 and so a(6) = 12-2*2 + 12-2*3 + 12-2*5 = 8 + 6 + 2 = 16. - _Wesley Ivan Hurt_, Mar 24 2018

%H Vincenzo Librandi, <a href="/A294013/b294013.txt">Table of n, a(n) for n = 1..2000</a>

%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(i).

%e 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.

%t Table[2 Sum[(n - i) (PrimePi[i] - PrimePi[i - 1]), {i, n}], {n, 40}]

%t Table[Total[#[[1]]-#[[2]]&/@Select[IntegerPartitions[2n,{2}],PrimeQ[ #[[2]]]&]],{n,60}] (* _Harvey P. Dale_, Jan 25 2019 *)

%Y Cf. A010051, A290288.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Oct 21 2017