A348983 a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

0, 1, 1, 7, 1, 14, 1, 31, 11, 20, 1, 80, 1, 26, 23, 111, 1, 109, 1, 122, 29, 38, 1, 328, 19, 44, 76, 164, 1, 250, 1, 351, 41, 56, 35, 565, 1, 62, 47, 514, 1, 334, 1, 248, 208, 74, 1, 1128, 27, 245, 59, 290, 1, 650, 47, 700, 65, 92, 1, 1336, 1, 98, 274, 1023, 53, 502, 1, 374, 77, 490, 1, 2213, 1, 116, 302, 416, 53

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A322582 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.

Dirichlet convolution of the identity function (A000027) with A348980.

Dirichlet convolution of sigma (A000203) with A348981.

Links

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

Crossrefs

Cf. A000005, A000027, A000203, A003958, A038040, A322582, A348980, A348981, A348982.

Cf. also A349123, A349143.

Programs

Mathematica

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

PARI

A038040(n) = (n*numdiv(n));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348983(n) = sumdiv(n,d,A038040(n/d)*A322582(d));

Formula

a(n) = Sum_{d|n} A038040(n/d) * A322582(d).

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

a(n) = Sum_{d|n} A000203(d) * A348981(n/d).

For all n >= 1, a(n) <= A349123(n) <= A349143(n).