A333645 a(n) = Sum_{d|n} uphi(d).

1, 2, 3, 5, 5, 6, 7, 12, 11, 10, 11, 15, 13, 14, 15, 27, 17, 22, 19, 25, 21, 22, 23, 36, 29, 26, 37, 35, 29, 30, 31, 58, 33, 34, 35, 55, 37, 38, 39, 60, 41, 42, 43, 55, 55, 46, 47, 81, 55, 58, 51, 65, 53, 74, 55, 84, 57, 58, 59, 75, 61, 62, 77, 121, 65, 66, 67, 85, 69, 70

Offset: 1

Author

Ilya Gutkovskiy, Mar 31 2020

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A047994.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117 (fixed points), A023900, A047994, A055653, A062949, A065464, A152649.

Programs

Mathematica

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[uphi[d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
A023900[n_] := Sum[MoebiusMu[d] d, {d, Divisors[n]}]; A062949[n_] := Sum[EulerPhi[d] DivisorSigma[0, d], {d, Divisors[n]}]; a[n_] := Sum[A023900[d] A062949[n/d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
f[p_,e_] := (p^(e+1) - e*p + e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Nov 12 2022 *)

PARI

uphi(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = sumdiv(n, d, uphi(d)); \\ Michel Marcus, Mar 31 2020

Formula

G.f.: Sum_{k>=1} uphi(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A023900(d) * A062949(n/d).

From Amiram Eldar, Nov 12 2022: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (2*p-1)/p^3) = A152649 * A065464 = 0.5793804872... . (End)