A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 5, 1, 9, 1, 17, 8, 13, 1, 37, 1, 17, 15, 49, 1, 51, 1, 57, 19, 25, 1, 117, 14, 29, 43, 77, 1, 105, 1, 129, 27, 37, 23, 191, 1, 41, 31, 185, 1, 141, 1, 117, 99, 49, 1, 325, 20, 117, 39, 137, 1, 237, 31, 253, 43, 61, 1, 405, 1, 65, 131, 321, 35, 213, 1, 177, 51, 209, 1, 579, 1, 77, 145, 197, 35, 249, 1, 521

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A322582 with the identity function, A000027.

Links

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

Crossrefs

Cf. A000027, A003958, A038040, A322582, A348981 (Möbius transform), A348982, A348983, A349130.

Cf. also A347130, A349140.

Programs

Mathematica

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*(n/# - s[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));
A348980(n) = sumdiv(n,d,d*A322582(n/d));

Formula

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

For all n >= 1, a(n) <= A347130(n) <= A349140(n).

a(n) = A038040(n) - A349130(n). - Antti Karttunen, Nov 14 2021