A344599 a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^3 - floor((n-1)/k)^3).

1, 8, 21, 46, 65, 126, 133, 224, 261, 364, 341, 618, 481, 722, 837, 1000, 833, 1404, 1045, 1718, 1641, 1798, 1541, 2760, 2065, 2516, 2673, 3346, 2465, 4410, 2821, 4256, 4041, 4312, 4469, 6462, 4033, 5390, 5637, 7504, 4961, 8532, 5461, 8186, 8613, 7906, 6533, 11736, 7861, 10640, 9621

Offset

1,2

Links

Table of n, a(n) for n=1..51.

Formula

Sum_{k=1..n} a(k) = A344522(n).

G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.

Mathematica

a[n_] := Sum[EulerPhi[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^3), {k, 1, n}]; Array[a, 40] (* Amiram Eldar, May 24 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, eulerphi(k)*((n\k)^3-((n-1)\k)^3));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3))

Crossrefs

Cf. A344522, A344598, A344600.

Sequence in context: A075629 A273602 A258448 * A138179 A067334 A066859

Adjacent sequences: A344596 A344597 A344598 * A344600 A344601 A344602

Keyword

nonn

Author

Seiichi Manyama, May 24 2021

Status

approved