A334940 Partial sums of A230595.

0, 0, 0, 1, 1, 3, 3, 3, 4, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 12, 14, 14, 14, 15, 17, 17, 17, 17, 17, 17, 17, 19, 21, 23, 23, 23, 25, 27, 27, 27, 27, 27, 27, 27, 29, 29, 29, 30, 30, 32, 32, 32, 32, 34, 34, 36, 38, 38, 38, 38, 40, 40, 40, 42, 42, 42, 42, 44, 44, 44, 44, 44, 46

Offset

1,6

Comments

Sum of the Dirichlet convolution of the characteristic function of primes (A010051) with itself from 1 to n.

(a(n) + A000720(floor(sqrt(n))))/2 equals the number of semiprimes <= n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} A010051(d) * A010051(k/d).

a(n) = 2*Sum_{p prime <= sqrt(n)} A000720(floor(n/p)) - A000720(floor(sqrt(n)))^2.

a(n) = 2*A072000(n) - A000720(floor(sqrt(n))).

a(n) = 2*A072613(n) + A000720(floor(sqrt(n))). - Vaclav Kotesovec, May 21 2020

a(n) ~ 2*n*log(log(n))/log(n). - Vaclav Kotesovec, May 21 2020

Maple

a:= proc(n) option remember; `if`(n<4, 0, a(n-1) +
     `if`(numtheory[bigomega](n)=2, `if`(issqr(n), 1, 2), 0))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 20 2020

Mathematica

f[n_] := DivisorSum[n, 1 &, PrimeQ[#] && PrimeQ[n/#] &]; Accumulate @ Array[f, 100] (* Amiram Eldar, May 20 2020 *)

Prog

(PARI) a(n) = my(s=sqrtint(n)); 2*sum(k=1, s, if(isprime(k), primepi(n\k), 0)) - primepi(s)^2;
(Python)
from math import isqrt
from sympy import primepi, prime
def A334940(n): return (int(sum(primepi(n//prime(k))-k+1 for k in range(1, primepi(isqrt(n))+1)))<<1) - primepi(isqrt(n)) # Chai Wah Wu, Jul 23 2024

Crossrefs

Cf. A000720, A001222, A010051, A072000, A230595.

Sequence in context: A361247 A151554 A221028 * A011978 A167025 A238505

Adjacent sequences: A334937 A334938 A334939 * A334941 A334942 A334943

Keyword

nonn

Author

Daniel Suteu, May 17 2020

Status

approved