A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).

1, 2, 7, 11, 23, 14, 47, 46, 70, 46, 119, 77, 167, 94, 161, 188, 287, 140, 359, 253, 329, 238, 527, 322, 596, 334, 642, 517, 839, 322, 959, 760, 833, 574, 1081, 770, 1367, 718, 1169, 1058, 1679, 658, 1847, 1309, 1610, 1054, 2207, 1316, 2346, 1192, 2009, 1837, 2807, 1284, 2737, 2162, 2513, 1678, 3479, 1771, 3719, 1918, 3290, 3056, 3841, 1666, 4487, 3157, 3689

Offset: 1

Author

Wesley Ivan Hurt, Jul 27 2025

Keywords

• nonn
• easy
• mult

Comments

Möbius transform of sigma(n) * phi(n) = A062354(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A062354, A330523.

Programs

Mathematica

Table[Sum[EulerPhi[d] DivisorSigma[1, d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]
f[p_, e_] := p^(2*e) - p^(e-1) - If[e > 1, p^(2*e-2) - p^(e-2), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 27 2025 *)

PARI

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i, 2];  p^(2*e) - p^(e - 1) - if(e > 1, p^(2*e - 2) - p^(e - 2), 1));} \\ Amiram Eldar, Jul 27 2025

Formula

a(n) = Sum_{d|n} A062354(d) * mu(n/d).

From Amiram Eldar, Jul 27 2025: (Start)

Multiplicative with a(p) = p^2 - 2, and a(p^e) = p^(2*e) - p^(2*e-2) - p^(e-1) + p^(e-2) for e >= 2.

Dirichlet g.f.: (zeta(s-2) * zeta(s-1) / zeta(s)) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s + 1/p^(2*s-2)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523*zeta(2)/(3*zeta(3)) = 0.24444595409976589792... . (End)