A349131 a(n) = Sum_{d|n} phi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and phi is Euler totient function.

1, 2, 4, 4, 8, 8, 12, 8, 14, 16, 20, 16, 24, 24, 32, 16, 32, 28, 36, 32, 48, 40, 44, 32, 52, 48, 46, 48, 56, 64, 60, 32, 80, 64, 96, 56, 72, 72, 96, 64, 80, 96, 84, 80, 112, 88, 92, 64, 114, 104, 128, 96, 104, 92, 160, 96, 144, 112, 116, 128, 120, 120, 168, 64, 192, 160, 132, 128, 176, 192, 140, 112, 144, 144, 208

Offset

1,2

Comments

Dirichlet convolution of A003958 with Euler totient function phi, A000010.

Möbius transform of A349130.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n} A000010(d) * A003958(n/d).

a(n) = Sum_{d|n} A008683(d) * A349130(n/d).

a(n) = Sum_{k=1..n} A003958(gcd(n, k)).

a(n) = A018804(n) - A348981(n).

For all n >= 1, a(n) <= A349171(n).

Multiplicative with a(p^e) = (p-1)*p^e - (p-2)*(p-1)^e. - Amiram Eldar, Nov 09 2021

Dirichlet g.f.: (zeta(s-1)/zeta(s)) / Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Dec 24 2023

Mathematica

f[p_, e_] := (p - 1)*p^e - (p - 2)*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

Prog

(PARI)
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349131(n) = sumdiv(n, d, eulerphi(d)*A003958(n/d));

Crossrefs

Cf. A000010, A003958, A018804, A348981, A349130 (inverse Möbius transform), A349132, A349171.

Sequence in context: A265322 A188112 A333194 * A366665 A166632 A116596

Adjacent sequences: A349128 A349129 A349130 * A349132 A349133 A349134

Keyword

nonn,mult

Author

Antti Karttunen, Nov 09 2021

Status

approved