A333465 - OEIS

%I #24 Mar 25 2020 08:02:09

%S 1,4,8,14,17,30,27,43,47,62,48,97,60,96,114,123,83,167,95,195,177,170,

%T 119,283,181,209,230,300,158,401,172,330,308,288,348,517,213,329,377,

%U 560,239,613,253,522,599,413,279,776,415,624,520,640,322,793,604,854,594,543,364,1220

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

%H Robert Israel, <a href="/A333465/b333465.txt">Table of n, a(n) for n = 1..5000</a>

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

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

%p f:= n -> add(ceil(n/k)*igcd(n,k),k=1..n):

%p map(f, [$1..100]); # _Robert Israel_, Mar 24 2020

%t Table[Sum[Ceiling[n/k] GCD[n, k], {k, n}], {n, 60}]

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

%o (PARI) a(n) = sum(k=1, n, ceil(n/k)*gcd(n, k)); \\ _Michel Marcus_, Mar 23 2020

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

%K nonn,look

%O 1,2

%A _Ilya Gutkovskiy_, Mar 22 2020