%I #24 Dec 26 2023 17:58:15
%S 0,0,0,2,5,5,10,8,15,12,12,12,23,18,31,31,31,24,41,41,60,60,60,60,83,
%T 72,72,72,72,59,88,88,119,119,119,119,119,102,139,139,139,120,161,161,
%U 204,204,204,204,251,228,228,228,228,228,281,281,281,281,281,281,340,311,372,372,372,341,341,341,408
%N a(n) = Sum_{i=floor((n-1)/2)..n-1} i * c(i), where c is the prime characteristic (A010051).
%C Sum of the primes in the n-th column of the example in A258087.
%H Robert Israel, <a href="/A288726/b288726.txt">Table of n, a(n) for n = 0..10000</a>
%F From _Robert Israel_, Jun 16 2017:
%F For k >= 2, a(2*k+1) - a(2*k) = 1-k if k-1 is prime, otherwise 0.
%F a(2*k+2) - a(2*k+1) = 2*k+1 if 2*k+1 is prime, otherwise 0. (End)
%F a(n) = A288656(n) - A288656(n-1), n>=1. - _Wesley Ivan Hurt_, Dec 26 2023
%p with(numtheory): A288726:=n->add(i*(pi(i)-pi(i-1)), i=floor((n-1)/2)..n-1): seq(A288726(n), n=0..100);
%p # Alternative:
%p M:= 100: # to get a(0) to a(2*M+1)
%p A:= Array(0..2*M+1):
%p A[3]:= 2:
%p for k from 2 to M do
%p if isprime(2*k-1) then A[2*k]:= A[2*k-1]+2*k-1 else A[2*k]:=A[2*k-1] fi;
%p if isprime(k-1) then A[2*k+1]:= A[2*k]-(k-1) else A[2*k+1]:= A[2*k] fi;
%p od:
%p convert(A,list); # _Robert Israel_, Jun 16 2017
%t Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, Floor[(n - 1)/2], n - 1}], {n, 0, 68}] (* _Michael De Vlieger_, Jun 14 2017 *)
%o (PARI) a(n) = sum(i=floor((n-1)/2), n-1, i*isprime(i)) \\ _Felix Fröhlich_, Jun 16 2017
%Y Cf. A010051, A258087, A288656.
%K nonn,easy
%O 0,4
%A _Wesley Ivan Hurt_, Jun 14 2017