A347134 -id:A347134 - cp's OEIS frontend

A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

0, 1, 1, 5, 1, 8, 1, 18, 8, 12, 1, 33, 1, 16, 14, 56, 1, 45, 1, 53, 18, 24, 1, 110, 14, 28, 45, 73, 1, 87, 1, 160, 26, 36, 22, 169, 1, 40, 30, 182, 1, 119, 1, 113, 93, 48, 1, 328, 20, 107, 38, 133, 1, 216, 30, 254, 42, 60, 1, 337, 1, 64, 125, 432, 34, 183, 1, 173, 50, 183, 1, 538, 1, 76, 135, 193, 34, 215, 1, 552, 216

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 with A003415.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Möbius transform of A347130.
Cf. A000010, A003415.
Cf. also A300251, A305809, A322577, A347132, A347133, A347134, A347235.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Sep 03 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347131(n) = sumdiv(n,d,A003415(n/d)*eulerphi(d));

A347131(n) = sum(k=1,n,A003415(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A003415(d).
a(n) = Sum_{d|n} A008683(n/d) * A347130(d).
a(n) = Sum_{k=1..n} A003415(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 6, 1, 2, 0, 16, 0, 2, 2, 24, 0, 19, 0, 20, 2, 2, 0, 72, 1, 2, 9, 24, 0, 40, 0, 80, 2, 2, 2, 111, 0, 2, 2, 96, 0, 48, 0, 32, 25, 2, 0, 256, 1, 29, 2, 36, 0, 117, 2, 120, 2, 2, 0, 244, 0, 2, 29, 240, 2, 64, 0, 44, 2, 56, 0, 446, 0, 2, 31, 48, 2, 72, 0, 352, 54, 2, 0, 308, 2, 2, 2, 168, 0, 304, 2, 56

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A003415 (arithmetic derivative) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A347131.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A003415, A069359, A347131.

Cf. also A300251, A305809, A347132, A347134, A347135.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347133(n) = sumdiv(n,d,A003415(n/d)*A069359(d));

a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A347131(d).

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

Original entry on oeis.org

0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A001615 (Dedekind psi function) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Dirichlet convolution

Cf. A001221, A001615, A069359, A322577.

Cf. also A347132, A347133, A347134.

Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));

a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A322577(d).

cp's OEIS Frontend

A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

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A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

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Formula

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula