A344485 a(n) = Sum_{d|n} (n-d) * phi(n/d).

0, 1, 4, 8, 16, 21, 36, 44, 60, 73, 100, 104, 144, 157, 180, 208, 256, 261, 324, 328, 376, 421, 484, 476, 560, 601, 648, 680, 784, 765, 900, 912, 984, 1057, 1108, 1128, 1296, 1333, 1396, 1420, 1600, 1569, 1764, 1768, 1836, 1981, 2116, 2064, 2268, 2305, 2436, 2504, 2704, 2673

Offset: 1

Author

Wesley Ivan Hurt, May 20 2021

Keywords

• nonn

Comments

a(n) is the sum of the (n - 1)-th antidiagonal in A074712. - Ctibor O. Zizka, Mar 14 2025

Möbius transform of A189835(n). - Wesley Ivan Hurt, Jul 16 2025

Examples

			a(6) = Sum_{d|6} (6-d) * phi(6/d) = 5*phi(6) + 4*phi(3) + 3*phi(2) + 0*phi(1) = 5*2 + 4*2 + 3*1 + 0*1 = 21.
		

Crossrefs

Cf. A000010, A000290, A018804, A074712, A189835.

Programs

Maple

with(numtheory): seq(add((n-d)*phi(n/d), d in divisors(n)), n=1..80); # Ridouane Oudra, Jan 21 2024

Mathematica

Table[Sum[(n - k)*EulerPhi[n/k^(1 - Ceiling[n/k] + Floor[n/k])] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

PARI

a(n) = sumdiv(n, d, (n-d) * eulerphi(n/d)); \\ Michel Marcus, May 21 2021

Formula

a(n) = A000290(n) - A018804(n). - Ridouane Oudra, Jan 21 2024

From Wesley Ivan Hurt, Jul 16 2025: (Start)

a(n) = Sum_{d|n} A189835(d) * mu(n/d).

a(p^k) = p^(2*k)-p^k-k*p^k+k*p^(k-1) for p prime and k>=1. (End)