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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.
7
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
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
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Nov 09 2021
STATUS
approved