A333450 -id:A333450 - cp's OEIS frontend

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

Ilya Gutkovskiy, Mar 21 2020

nonn

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

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}]

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

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).

A333471 a(n) = 2 * mu(n) + Sum_{d|n, d > 1} mu(n/d) * (prime(d) - prime(d-1)).

Original entry on oeis.org

2, -1, 0, 1, 2, 1, 2, 0, 2, 3, 0, 3, 2, -1, 0, 4, 4, -2, 4, -3, -2, 5, 2, 0, 4, 1, -2, 1, 0, -3, 12, -2, 4, -3, 4, -4, 4, 1, 0, 2, 4, 1, 8, -5, -2, -1, 10, 2, 0, -8, -2, 1, 0, 10, 2, 2, 0, 1, 4, -1, 0, -3, 10, 0, -4, -7, 12, 3, 6, -9, 2, 4, 6, 1, -2, -3, 2, 3, 2, -2

Offset: 1

Ilya Gutkovskiy, Mar 23 2020

sign

Moebius transform of A054541 (2 followed by prime gaps).

Cf. A000040, A001223, A007444, A008683, A030013, A054541, A182986, A333450 (partial sums).

a[n_] := 2 MoebiusMu[n] + Sum[If[d > 1, MoebiusMu[n/d] (Prime[d] - Prime[d - 1]), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 80}]

a(n) = Sum_{d|n} mu(n/d) * A054541(d).

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

cp's OEIS Frontend

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

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