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

1,3

Comments

If p is prime, a(p) = Sum_{d|p} (p-d) * phi(p/d) = (p-1) * phi(p) + (p-p) * phi(1) = (p-1)^2.

Links

Table of n, a(n) for n=1..54.

Formula

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

Example

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.

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}]

Prog

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

Crossrefs

Cf. A000010, A000290, A018804.

Sequence in context: A312808 A249486 A129370 * A212009 A312809 A312810

Adjacent sequences: A344482 A344483 A344484 * A344486 A344487 A344488

Keyword

nonn

Author

Wesley Ivan Hurt, May 20 2021

Status

approved