A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 4, 1, 7, 1, 12, 7, 11, 1, 24, 1, 15, 13, 32, 1, 35, 1, 40, 17, 23, 1, 68, 13, 27, 35, 56, 1, 71, 1, 80, 25, 35, 21, 112, 1, 39, 29, 116, 1, 99, 1, 88, 77, 47, 1, 176, 19, 91, 37, 104, 1, 151, 29, 164, 41, 59, 1, 232, 1, 63, 105, 192, 33, 155, 1, 136, 49, 159, 1, 308, 1, 75, 117, 152, 33, 183, 1, 304, 151

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of Euler phi (A000010) with A322582.

Möbius transform of A348980.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000010, A003958, A008683, A018804, A322582, A348980 (Inverse Möbius transform), A348981, A348982, A348983, A349131.

Cf. also A347131, A349141.

Programs

Mathematica

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#]) * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

PARI

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348981(n) = sumdiv(n,d,A322582(n/d)*eulerphi(d));

Formula

a(n) = Sum_{d|n} A000010(n/d) * A322582(d).

a(n) = Sum_{d|n} A008683(n/d) * A348980(d).

a(n) = Sum_{k=1..n} A322582(gcd(n,k)).

For all n >= 1, a(n) <= A347131(n) <= A349141(n).

a(n) = A018804(n) - A349131(n). - Antti Karttunen, Nov 14 2021