A288726 a(n) = Sum_{i=floor((n-1)/2)..n-1} i * c(i), where c is the prime characteristic (A010051).

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, 72, 72, 72, 72, 59, 88, 88, 119, 119, 119, 119, 119, 102, 139, 139, 139, 120, 161, 161, 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

Offset: 0

Author

Wesley Ivan Hurt, Jun 14 2017

Keywords

• nonn
• easy

Comments

Sum of the primes in the n-th column of the example in A258087.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A010051, A258087, A288656.

Programs

Maple

with(numtheory): A288726:=n->add(i*(pi(i)-pi(i-1)), i=floor((n-1)/2)..n-1): seq(A288726(n), n=0..100);
# Alternative:
M:= 100: # to get a(0) to a(2*M+1)
A:= Array(0..2*M+1):
A[3]:= 2:
for k from 2 to M do
   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;
   if isprime(k-1) then A[2*k+1]:= A[2*k]-(k-1) else A[2*k+1]:= A[2*k] fi;
od:
convert(A,list); # Robert Israel, Jun 16 2017

Mathematica

Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, Floor[(n - 1)/2], n - 1}], {n, 0, 68}] (* Michael De Vlieger, Jun 14 2017 *)

PARI

a(n) = sum(i=floor((n-1)/2), n-1, i*isprime(i)) \\ Felix Fröhlich, Jun 16 2017

Formula

From Robert Israel, Jun 16 2017: (Start)

For k >= 2, a(2*k+1) - a(2*k) = 1-k if k-1 is prime, otherwise 0.

a(2*k+2) - a(2*k+1) = 2*k+1 if 2*k+1 is prime, otherwise 0. (End)

a(n) = A288656(n) - A288656(n-1), n>=1. - Wesley Ivan Hurt, Dec 26 2023