A003966 -id:A003966 - cp's OEIS frontend

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

Antti Karttunen, Nov 09 2021

Dirichlet convolution of A003958 with the identity function, A000027.

Dirichlet convolution of sigma (A000203) with A003966.

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

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

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 *)

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

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

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

Original entry on oeis.org

1, -2, -2, -2, -2, 4, -2, -2, -4, 4, -2, 4, -2, 4, 4, -2, -2, 8, -2, 4, 4, 4, -2, 4, -8, 4, -8, 4, -2, -8, -2, -2, 4, 4, 4, 8, -2, 4, 4, 4, -2, -8, -2, 4, 8, 4, -2, 4, -12, 16, 4, 4, -2, 16, 4, 4, 4, 4, -2, -8, -2, 4, 8, -2, 4, -8, -2, 4, 4, -8, -2, 8, -2, 4, 16, 4, 4, -8, -2, 4, -16, 4, -2, -8, 4, 4, 4, 4, -2, -16

Offset: 1

Antti Karttunen, Nov 16 2021

Multiplicative because both A003958 and A063441 are.

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003959 with factor A003958. For example, convolving this with A003968 (the Möbius transform of A003959) produces A003966, the Möbius transform of A003958.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Wikipedia, Dirichlet convolution

Cf. A003958, A003959, A003966, A003968, A063441, A349356 (Dirichlet inverse), A349357 (sum with it).

Cf. also A349382.

f[p_, e_] := -2*(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349355(n) = sumdiv(n,d,A003958(n/d)*A063441(d));

a(n) = Sum_{d|n} A003958(n/d) * A063441(d).

Multiplicative with a(p^e) = -2*(p-1)^(e-1). - Amiram Eldar, Nov 16 2021

cp's OEIS Frontend

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

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