cp's OEIS Frontend

A294023 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part prime.

%I A294023 #32 Nov 25 2019 02:57:22
%S A294023 0,0,0,0,1,2,4,6,8,10,13,16,19,22,26,30,34,38,42,46,50,54,59,64,69,74,
%T A294023 80,86,92,98,104,110,116,122,129,136,143,150,158,166,174,182,190,198,
%U A294023 206,214,223,232,241,250,259,268,277,286,295,304,313,322,332
%N A294023 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part prime.
%C A294023 Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 at prime values of x for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(11), x=2,3,5 and so 11-2*2 + 11-2*3 + 11-2*5 = 7 + 5 + 1 = 13. - _Wesley Ivan Hurt_, Mar 24 2018
%H A294023 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A294023 a(n) = Sum_{i=1..floor(n/2)} (n - 2i)*A010051(i).
%F A294023 First differences are A056172. - _David A. Corneth_, Apr 06 2018
%F A294023 a(n) = Sum_{i=1..n-1} pi(floor(i/2)), where pi(n) = A000720(n). - _Ridouane Oudra_, Nov 24 2019
%e A294023 The partitions of n = 11 into a number and a smaller prime number are 9 + 2, 8 + 3, and 6 + 5, so a(11) = (9 - 2) + (8 - 3) + (6 - 5) = 13. - _Michael B. Porter_, Apr 06 2018
%p A294023 with(numtheory): seq(add(pi(floor(i/2)), i=1..n-1), n=1..100); # _Ridouane Oudra_, Nov 24 2019
%t A294023 Table[Sum[(n - 2 i) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 60}]
%o A294023 (PARI) a(n) = sum(i=1, n\2, (n - 2*i)*isprime(i)); \\ _Michel Marcus_, Mar 24 2018
%o A294023 (PARI) a(n) = my(res = 0); forprime(p = 2, n >> 1, res += (n - p << 1)); res \\ _David A. Corneth_, Apr 06 2018
%Y A294023 Cf. A010051, A056172, A294022, A000720.
%K A294023 nonn,easy
%O A294023 1,6
%A A294023 _Wesley Ivan Hurt_, Oct 21 2017