A333839 a(n) = Sum_{k = 4..n} ((prevprime(k) + nextprime(k))/2 - k).

0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 2, 4, 5, 5, 4, 2, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 0, 0, 2, 4, 5, 5, 4, 2, 1, 2, 2, 1, 2, 4, 5, 5, 4, 2, 3, 6, 8, 9, 9, 8, 6

Offset: 4

Author

Ctibor O. Zizka, Apr 07 2020

Keywords

• nonn

Comments

It looks like a(n) >= 0 for all n >= 4.

If (n,n+2) are twin primes, then a(n) = 0 and a(n+1) = 0.

Partial sums of b(k) = prevprime(k) + nextprime(k) - 2*k; b(k) = 0 for A145025.

Examples

			a(4) = (3 + 5)/2 - 4 = 0;
a(5) = (3 + 5)/2 - 4 + (3 + 7)/2 - 5 = 0.
		

Crossrefs

Cf. A013632, A033559, A049711, A145025.

Programs

Mathematica

Array[Sum[Total[NextPrime[k, {-1, 1}]]/2 - k, {k, 4, #}] &, 92, 4] (* Michael De Vlieger, Apr 10 2020 *)

Formula

a(n) = Sum_{k = 4..n} (A013632(k) - A049711(k)) / 2.