cp's OEIS Frontend

A294022 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part prime.

%I A294022 #29 Apr 16 2018 18:59:15
%S A294022 0,0,1,2,1,4,3,8,6,4,3,12,10,20,18,16,14,28,25,40,36,32,29,48,44,40,
%T A294022 37,34,31,56,52,78,73,68,64,60,56,88,84,80,76,112,107,144,138,132,127,
%U A294022 168,162,156,150,144,138,184,177,170,163,156,150,202,195,248
%N A294022 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part prime.
%C A294022 Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 such that n-x is prime for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(12), the integer values of x which make 12-x prime are x=1,5 and so a(12) = 12-2*1 + 12-2*5 = 10 + 2 = 12. - _Wesley Ivan Hurt_, Mar 24 2018
%H A294022 Robert G. Wilson v, <a href="/A294022/b294022.txt">Table of n, a(n) for n = 1..1000</a>
%H A294022 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A294022 a(n) = Sum_{i=1..floor(n/2)} (n - 2i) * A010051(n - i).
%e A294022 There are two ways to partition n = 9 into a prime and a smaller positive integer: 7 + 2 and 5 + 4. So a(9) = (7 - 2) + (5 - 4) = 6. - _Michael B. Porter_, Mar 26 2018
%t A294022 Table[Sum[(n - 2 i) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[n/2]}], {n, 60}]
%o A294022 (PARI) a(n) = sum(i=1, n\2, (n - 2*i)*isprime(n-i)); \\ _Michel Marcus_, Mar 24 2018
%Y A294022 Cf. A010051, A294023.
%K A294022 nonn,easy
%O A294022 1,4
%A A294022 _Wesley Ivan Hurt_, Oct 21 2017