A347135 -id:A347135 - cp's OEIS frontend

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

1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660

Offset: 1

Ilya Gutkovskiy, Aug 29 2019

Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
Dirichlet convolution of A008966 with A018804.
Dirichlet convolution of A038040 with A271102.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Dedekind Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A001615, A008966, A018804, A038040, A271102.

Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.

f:= proc(n) local t;
  mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019

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)));
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022

Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
a(p) = 2*p, where p is prime.
Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019
a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019
a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 30, 10, 16, 1, 65, 1, 20, 18, 104, 1, 83, 1, 93, 22, 28, 1, 254, 16, 32, 63, 121, 1, 167, 1, 320, 30, 40, 26, 391, 1, 44, 34, 374, 1, 215, 1, 177, 143, 52, 1, 840, 22, 165, 42, 205, 1, 450, 34, 494, 46, 64, 1, 827, 1, 68, 183, 912, 38, 311, 1, 261, 54, 295, 1, 1430, 1, 80, 197, 289, 38, 359

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).

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

Cf. A001615, A003415.

Cf. also A305809, A322577, A347131, A347133, A347135.

Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* Michael De Vlieger, Oct 19 2021, after Jan Mangaldan at A001615 *)

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));

a(n) = Sum_{d|n} A001615(n/d) * A003415(d).

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).

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 8, 5, 12, 1, 23, 1, 16, 14, 20, 1, 36, 1, 35, 18, 24, 1, 60, 9, 28, 21, 47, 1, 87, 1, 48, 26, 36, 22, 103, 1, 40, 30, 92, 1, 119, 1, 71, 66, 48, 1, 148, 13, 92, 38, 83, 1, 144, 30, 124, 42, 60, 1, 247, 1, 64, 86, 112, 34, 183, 1, 107, 50, 183, 1, 268, 1, 76, 110, 119, 34, 215, 1, 228, 81, 84

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 (Euler totient function phi) with A069359.

Dirichlet convolution of A001221 (omega) with A029935 (the convolution square of Euler phi).

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

Cf. A000010, A001221, A069359, A029935.

Cf. also A347131, A347133, A347135.

Table[DivisorSum[n,EulerPhi[n/#]*#*DivisorSum[#,1/#&,PrimeQ]&],{n,82}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347134(n) = sumdiv(n,d,eulerphi(n/d)*A069359(d));

a(n) = Sum_{d|n} A000010(n/d) * A069359(d)
a(n) = Sum_{d|n} A001221(n/d) * A029935(d).
a(n) = Sum_{k=1..n} A069359(gcd(n,k)). - Antti Karttunen, Oct 17 2021

cp's OEIS Frontend

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

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A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

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

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Author

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Programs

PARI

Formula

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

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Author

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Programs

Mathematica

PARI

Formula