A333465 a(n) = Sum_{k=1..n} ceiling(n/k) * gcd(n,k).

1, 4, 8, 14, 17, 30, 27, 43, 47, 62, 48, 97, 60, 96, 114, 123, 83, 167, 95, 195, 177, 170, 119, 283, 181, 209, 230, 300, 158, 401, 172, 330, 308, 288, 348, 517, 213, 329, 377, 560, 239, 613, 253, 522, 599, 413, 279, 776, 415, 624, 520, 640, 322, 793, 604, 854, 594, 543, 364, 1220

Offset: 1

Author

Ilya Gutkovskiy, Mar 22 2020

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000005, A000010, A006590, A018804, A333463.

Programs

Maple

f:= n -> add(ceil(n/k)*igcd(n,k),k=1..n):
map(f, [$1..100]); # Robert Israel, Mar 24 2020

Mathematica

Table[Sum[Ceiling[n/k] GCD[n, k], {k, n}], {n, 60}]
Table[Sum[EulerPhi[n/d] (d + Sum[DivisorSigma[0, k], {k, d - 1}]), {d, Divisors[n]}], {n, 60}]

PARI

a(n) = sum(k=1, n, ceil(n/k)*gcd(n, k)); \\ Michel Marcus, Mar 23 2020

Formula

a(n) = Sum_{d|n} phi(n/d) * A006590(d).

a(n) = A018804(n) + Sum_{k=1..n-1} Sum_{d|k} gcd(n,d).