A348011 - OEIS

%I #10 Nov 05 2022 08:18:25

%S 1,4,10,20,28,40,54,88,102,112,130,200,180,216,280,368,304,408,378,

%T 560,540,520,550,880,740,720,954,1080,868,1120,990,1504,1300,1216,

%U 1512,2040,1404,1512,1800,2464,1720,2160,1890,2600,2856,2200,2254,3680,2730,2960

%N a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

%F Multiplicative with a(p^e) = p^(e-1) * ((p + 1) * p^e - 2).

%F a(n) = Sum_{k=1..n, gcd(n,k) = 1} gcd(n,k-1)^2.

%F a(n) = Sum_{k=1..n} uphi(gcd(n,k)^2).

%F a(n) = Sum_{d|n} phi(n/d) * uphi(d^2).

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.4083249979... . - _Amiram Eldar_, Nov 05 2022

%t Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]

%t f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]

%o (PARI) a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ _Michel Marcus_, Sep 24 2021

%Y Cf. A000010, A001221, A002618, A007434, A034444, A047994, A060648, A062355, A069097, A191414, A341772.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Sep 24 2021