OFFSET
0,4
COMMENTS
Partial sums of A069359.
LINKS
Wikipedia, Prime Zeta Function
FORMULA
a(n) ~ A085548 * n*(n+1)/2.
a(n) = Sum_{p prime <= n} A000217(floor(n/p)).
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p)*floor(1+n/p))/2, where pi(x) is the prime-counting function (A000720).
a(n) = Sum_{i=1..n} i*pi(floor(n/i)), where pi(n) = A000720(n). - Ridouane Oudra, Oct 16 2019
MAPLE
with(numtheory): seq(add(i*pi(floor(n/i)), i=1..n), n=0..60); # Ridouane Oudra, Oct 16 2019
MATHEMATICA
a[n_] := Module[{s=0, p=2}, While[p<=n, s += (Floor[n/p] * Floor[1 + n/p]); p=NextPrime[p]]; s]/2; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2018 *)
PROG
(PARI) a(n) = my(s=0); forprime(p=2, n, s+=(n\p)*(1+n\p)); s/2;
(PARI) a(n) = sum(k=1, sqrtint(n), k*(k+1) * (primepi(n\k) - primepi(n\(k+1))))/2 + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), (n\k)*(1+n\k), 0))/2;
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Suteu, Nov 25 2018
STATUS
approved