A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

1, 4, 10, 20, 28, 40, 54, 88, 102, 112, 130, 200, 180, 216, 280, 368, 304, 408, 378, 560, 540, 520, 550, 880, 740, 720, 954, 1080, 868, 1120, 990, 1504, 1300, 1216, 1512, 2040, 1404, 1512, 1800, 2464, 1720, 2160, 1890, 2600, 2856, 2200, 2254, 3680, 2730, 2960

Offset

1,2

Links

Table of n, a(n) for n=1..50.

Formula

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

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

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

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

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

Mathematica

Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]
f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]

Prog

(PARI) a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ Michel Marcus, Sep 24 2021

Crossrefs

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

Sequence in context: A145021 A135280 A100436 * A009882 A100439 A301193

Adjacent sequences: A348008 A348009 A348010 * A348012 A348013 A348014

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Sep 24 2021

Status

approved