A333450 a(n) = Sum_{k=1..n} mu(k) * prime(floor(n/k)).

2, 1, 1, 2, 4, 5, 7, 7, 9, 12, 12, 15, 17, 16, 16, 20, 24, 22, 26, 23, 21, 26, 28, 28, 32, 33, 31, 32, 32, 29, 41, 39, 43, 40, 44, 40, 44, 45, 45, 47, 51, 52, 60, 55, 53, 52, 62, 64, 64, 56, 54, 55, 55, 65, 67, 69, 69, 70, 74, 73, 73, 70, 80, 80, 76, 69, 81, 84, 90, 81, 83, 87, 93, 94

Offset: 1

Author

Ilya Gutkovskiy, Mar 21 2020

Keywords

• nonn

Crossrefs

Cf. A000040, A001223, A007444, A007504, A008683, A333449.

Programs

Mathematica

Table[Sum[MoebiusMu[k] Prime[Floor[n/k]], {k, 1, n}], {n, 1, 74}]
g[1] = 2; g[n_] := Prime[n] - Prime[n - 1]; a[n_] := Sum[Sum[MoebiusMu[k/d] g[d], {d, Divisors[k]}], {k, 1, n}]; Table[a[n], {n, 1, 74}]

PARI

a(n) = sum(k=1, n, moebius(k)*prime(n\k)); \\ Michel Marcus, Mar 22 2020

Python

from functools import lru_cache
from sympy import prime
@lru_cache(maxsize=None)
def A333450(n):
    c, j = 2*(n+1)-prime(n), 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A333450(k1)
        j, k1 = j2, n//j2
    return 2*j-c # Chai Wah Wu, Mar 31 2021

Formula

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