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

1,4

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

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).

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 *)

Prog

(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));

Crossrefs

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

Cf. also A349123, A349143.

Sequence in context: A268919 A040055 A317016 * A013614 A179530 A098081

Adjacent sequences: A348980 A348981 A348982 * A348984 A348985 A348986

Keyword

nonn

Author

Antti Karttunen, Nov 08 2021

Status

approved