A322068 a(n) = (1/2)*Sum_{p prime <= n} floor(n/p) * floor(1 + n/p).
0, 0, 1, 2, 4, 5, 10, 11, 15, 18, 25, 26, 36, 37, 46, 54, 62, 63, 78, 79, 93, 103, 116, 117, 137, 142, 157, 166, 184, 185, 216, 217, 233, 247, 266, 278, 308, 309, 330, 346, 374, 375, 416, 417, 443, 467, 492, 493, 533, 540, 575, 595, 625, 626, 671, 687, 723, 745
Offset: 0
Links
- Wikipedia, Prime Zeta Function
Programs
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Maple
with(numtheory): seq(add(i*pi(floor(n/i)), i=1..n), n=0..60); # Ridouane Oudra, Oct 16 2019
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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 *) -
PARI
a(n) = my(s=0); forprime(p=2, n, s+=(n\p)*(1+n\p)); s/2;
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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;
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
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