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

0,4

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

Formula

From Robert Israel, Jun 16 2017:

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

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 *)

Prog

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

Crossrefs

Cf. A010051, A258087, A288656.

Sequence in context: A243333 A059797 A173567 * A344572 A265129 A212624

Adjacent sequences: A288723 A288724 A288725 * A288727 A288728 A288729

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 14 2017

Status

approved