A349130 a(n) = Sum_{d|n} d * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1).

1, 3, 5, 7, 9, 15, 13, 15, 19, 27, 21, 35, 25, 39, 45, 31, 33, 57, 37, 63, 65, 63, 45, 75, 61, 75, 65, 91, 57, 135, 61, 63, 105, 99, 117, 133, 73, 111, 125, 135, 81, 195, 85, 147, 171, 135, 93, 155, 127, 183, 165, 175, 105, 195, 189, 195, 185, 171, 117, 315, 121, 183, 247, 127, 225, 315, 133, 231, 225, 351, 141

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A003958 with the identity function, A000027.

Dirichlet convolution of sigma (A000203) with A003966.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000203, A003958, A003966, A038040, A348980, A349129, A349131 (Möbius transform), A349132, A349133, A349170.

Programs

Mathematica

f[p_, e_] := p^(e + 1) - (p - 1)^(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)

PARI

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349130(n) = sumdiv(n,d,d*A003958(n/d));

Formula

a(n) = Sum_{d|n} d * A003958(n/d).

a(n) = Sum_{d|n} A349131(d).

a(n) = Sum_{d|n} A000203(d) * A003966(n/d).

a(n) = A038040(n) - A348980(n).

For all n >= 1, a(n) <= A349129(n) <= A349170(n).

Multiplicative with a(p^e) = p^(e+1) - (p-1)^(e+1). - Amiram Eldar, Nov 09 2021