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a(n) = Sum_{d|n} lcm(d, n/d) / d.
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%I #13 Dec 05 2022 04:46:14

%S 1,3,4,6,6,12,8,12,11,18,12,24,14,24,24,23,18,33,20,36,32,36,24,48,27,

%T 42,32,48,30,72,32,45,48,54,48,66,38,60,56,72,42,96,44,72,66,72,48,92,

%U 51,81,72,84,54,96,72,96,80,90,60,144,62,96,88,88,84,144,68,108,96,144

%N a(n) = Sum_{d|n} lcm(d, n/d) / d.

%H Antti Karttunen, <a href="/A332619/b332619.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Ga#gcd">Index entries for sequences related to gcd's</a>.

%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.

%F a(n) = Sum_{d|n} d / gcd(d, n/d).

%F From _Amiram Eldar_, Dec 05 2022: (Start)

%F Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) + e/2 if e is even, and (p^(e+2)-p)/(p^2-1) + (e + 1)/2 if e is odd.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*zeta(6)/(8*zeta(5)) = 0.740543... . (End)

%p a:= n-> add(d/igcd(d, n/d), d=numtheory[divisors](n)):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Feb 17 2020

%t Table[Sum[LCM[d, n/d]/d, {d, Divisors[n]}], {n, 1, 70}]

%t f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1) + e/2, (p^(e + 2) - p)/(p^2 - 1) + (e + 1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 05 2022 *)

%o (PARI) A332619(n) = sumdiv(n,d,lcm(d,n/d)/d); \\ _Antti Karttunen_, Nov 12 2021

%Y Cf. A055155, A057660, A057661, A057670, A332618.

%Y Cf. A013663, A013664.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 17 2020