A348982 a(n) = Sum_{d|n} psi(n/d) * A322582(d), where psi is Dedekind psi (A001615), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 6, 1, 11, 1, 22, 9, 15, 1, 52, 1, 19, 17, 66, 1, 69, 1, 76, 21, 27, 1, 176, 15, 31, 51, 100, 1, 145, 1, 178, 29, 39, 25, 288, 1, 43, 33, 264, 1, 189, 1, 148, 123, 51, 1, 508, 21, 145, 41, 172, 1, 339, 33, 352, 45, 63, 1, 632, 1, 67, 159, 450, 37, 277, 1, 220, 53, 265, 1, 924, 1, 79, 175, 244, 37, 321

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A001615 with A322582.

Links

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

Crossrefs

Cf. A001615, A003958, A322582, A327251, A348980, A348981, A348982, A348983, A349132.

Cf. also A347132, A349142.

Programs

Mathematica

f1[p_, e_] := (p + 1)*p^(e - 1); psi[1] = 1; psi[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*psi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348982(n) = sumdiv(n,d,A001615(n/d)*A322582(d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A322582(d).

For all n >= 1, a(n) <= A347132(n) <= A349142(n).

a(n) = A327251(n) - A349132(n). - Antti Karttunen, Nov 14 2021