A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

0, 1, 1, 7, 1, 11, 1, 33, 10, 15, 1, 61, 1, 19, 17, 131, 1, 77, 1, 89, 21, 27, 1, 263, 16, 31, 67, 117, 1, 145, 1, 473, 29, 39, 25, 379, 1, 43, 33, 395, 1, 189, 1, 173, 137, 51, 1, 997, 22, 155, 41, 201, 1, 443, 33, 527, 45, 63, 1, 743, 1, 67, 177, 1611, 37, 277, 1, 257, 53, 265, 1, 1541, 1, 79, 187, 285, 37, 321

Offset: 1

Author

Antti Karttunen, Nov 08 2021

Keywords

• nonn

Comments

Dirichlet convolution of A348507 with the identity function, A000027.

Dirichlet convolution of sigma with A348971.

Links

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

Crossrefs

Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.

Cf. also A347130, A348980.

Programs

Mathematica

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

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349140(n) = sumdiv(n,d,d*A348507(n/d));

Formula

a(n) = Sum_{d|n} d * A348507(n/d).

a(n) = Sum_{d|n} A000203(d) * A348971(n/d).

a(n) = Sum_{d|n} A349141(d).

For all n >= 1, a(n) >= A347130(n) >= A348980(n).

a(n) = A349170(n) - A038040(n). - Antti Karttunen, Nov 15 2021