A349131 - OEIS

%I #16 Dec 24 2023 01:09:08

%S 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,

%T 48,46,48,56,64,60,32,80,64,96,56,72,72,96,64,80,96,84,80,112,88,92,

%U 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

%N 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.

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

%C Möbius transform of A349130.

%H Antti Karttunen, <a href="/A349131/b349131.txt">Table of n, a(n) for n = 1..20000</a>

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

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

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

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

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

%F Multiplicative with a(p^e) = (p-1)*p^e - (p-2)*(p-1)^e. - _Amiram Eldar_, Nov 09 2021

%F Dirichlet g.f.: (zeta(s-1)/zeta(s)) / Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - _Amiram Eldar_, Dec 24 2023

%t 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 *)

%o (PARI)

%o A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };

%o A349131(n) = sumdiv(n,d,eulerphi(d)*A003958(n/d));

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

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Nov 09 2021