A333449 a(n) = Sum_{k=1..n} prime(floor(n/k)).

2, 5, 9, 14, 20, 27, 33, 40, 48, 61, 65, 80, 86, 95, 107, 120, 128, 141, 149, 168, 178, 189, 195, 218, 232, 243, 253, 268, 272, 297, 313, 330, 342, 353, 373, 396, 404, 419, 431, 458, 466, 483, 495, 510, 530, 539, 553, 594, 604, 627, 641, 660, 664, 689, 703, 726, 742, 749, 757, 798

Offset

1,1

Links

Table of n, a(n) for n=1..60.

Formula

G.f.: (1/(1 - x)) * (2*x/(1 - x) + Sum_{k>=2} (prime(k) - prime(k-1))*x^k/(1 - x^k)).

Sum_{k=1..n} mu(k) * a(floor(n/k)) = prime(n).

Mathematica

Table[Sum[Prime[Floor[n/k]], {k, 1, n}], {n, 1, 60}]
g[1] = 2; g[n_] := Prime[n] - Prime[n - 1]; a[n_] := Sum[Sum[g[d], {d, Divisors[k]}], {k, 1, n}]; Table[a[n], {n, 1, 60}]

Prog

(PARI) a(n) = sum(k=1, n, prime(n\k)); \\ Michel Marcus, Mar 22 2020
(Python)
from sympy import prime
def A333449(n):
    c, j = 0, 1
    while j <= n:
        c += prime(k:=n//j)*(-j+(j:=n//k+1))
    return c # Chai Wah Wu, Jan 29 2026

Crossrefs

Cf. A000040, A001223, A007445, A007504, A008683, A333450.

Sequence in context: A361742 A132296 A275586 * A075543 A132315 A079509

Adjacent sequences: A333446 A333447 A333448 * A333450 A333451 A333452

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 21 2020

Status

approved