A333576 a(1) = 1; thereafter a(n) = n * uphi(n) / 2.

1, 1, 3, 6, 10, 6, 21, 28, 36, 20, 55, 36, 78, 42, 60, 120, 136, 72, 171, 120, 126, 110, 253, 168, 300, 156, 351, 252, 406, 120, 465, 496, 330, 272, 420, 432, 666, 342, 468, 560, 820, 252, 903, 660, 720, 506, 1081, 720, 1176, 600, 816, 936, 1378, 702, 1100, 1176, 1026, 812, 1711, 720

Offset: 1

Author

Ilya Gutkovskiy, Mar 27 2020

Keywords

• nonn
• easy

Comments

The unitary version of A023896.

Crossrefs

Cf. A001221, A023896, A047994, A065464, A076479, A145388, A353908.

Programs

Mathematica

uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[1] = 1; a[n_] := n uphi[n]/2; Table[a[n], {n, 1, 60}]
a[n_] := (n/2) Sum[If[GCD[d, n/d] == 1, (-1)^PrimeNu[n/d] (d + 1), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

PARI

a(n) = if(n == 1, 1, my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - 1) * n / 2); \\ Amiram Eldar, Sep 21 2024

Formula

a(n) = (n/2) * Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * (d + 1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/36) * Product_{p prime} (1 - (2*p-1)/p^3) = A353908 * A065464 = 0.117407... . - Amiram Eldar, Sep 21 2024