A319998 -id:A319998 - cp's OEIS frontend

A319997 a(n) = Sum_{d|n, d is odd} mu(n/d)*d, where mu(n) is Moebius function A008683.

1, -1, 2, 0, 4, -2, 6, 0, 6, -4, 10, 0, 12, -6, 8, 0, 16, -6, 18, 0, 12, -10, 22, 0, 20, -12, 18, 0, 28, -8, 30, 0, 20, -16, 24, 0, 36, -18, 24, 0, 40, -12, 42, 0, 24, -22, 46, 0, 42, -20, 32, 0, 52, -18, 40, 0, 36, -28, 58, 0, 60, -30, 36, 0, 48, -20, 66, 0, 44, -24, 70, 0, 72, -36, 40, 0, 60, -24, 78, 0, 54, -40, 82, 0, 64, -42, 56, 0, 88

Offset: 1

Antti Karttunen, Oct 31 2018

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

Cf. A000010, A000035, A008683, A062570, A319998.

A319997(n) = sumdiv(n,d,(d%2)*moebius(n/d)*d);

A319997(n) = if(n%2, eulerphi(n), if(n%4, -eulerphi(n), 0));

A319997(n) =  { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],-(1==f[i,2]),(f[i,1]-1)*(f[i,1]^(f[i,2]-1)))); };

a(n) = Sum_{d|n} A000035(d)*A008683(n/d)*d.
a(n) = A000010(n) - A319998(n).
For even n, a(n) = A000010(n) - 2*A000010(n/2); for odd n, a(n) = A000010(n).
a(2n+1) = A000010(2n+1), a(4n+2) = -A000010(4n+2), a(4n) = 0.
Multiplicative with a(2^1) = -1, a(2^e) = 0 for e > 1, and a(p^e) = (p - 1)*p^(e-1) when p is an odd prime.
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Nov 02 2018
Dirichlet g.f.: zeta(s-1)*(1-2^(1-s))/zeta(s). - R. J. Mathar, Jan 07 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - Amiram Eldar, Nov 12 2022

A323237 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291756(n) for all n, except f(1) = -1 and for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 15, 3, 23, 3, 24, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 36, 3, 40, 3, 41, 3, 27, 3, 42, 3, 43, 3, 44, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3, 50, 3, 51, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:

A319701(i) = A319701(j) => a(i) = a(j) => A319998(i) = A319998(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A173557, A291756, A295887, A319701, A319998, A322587, A323238, A323241.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
Aux323237(n) = if(1==n,-1,if(n%2,0,(1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n))));
v323237 = rgs_transform(vector(up_to, n, Aux323237(n)));
A323237(n) = v323237[n];

A325596 a(n) = Sum_{d|n} mu(n/d) * (-1)^(d + 1) * d.

Original entry on oeis.org

1, -3, 2, -2, 4, -6, 6, -4, 6, -12, 10, -4, 12, -18, 8, -8, 16, -18, 18, -8, 12, -30, 22, -8, 20, -36, 18, -12, 28, -24, 30, -16, 20, -48, 24, -12, 36, -54, 24, -16, 40, -36, 42, -20, 24, -66, 46, -16, 42, -60, 32, -24, 52, -54, 40, -24, 36, -84, 58, -16, 60, -90, 36, -32, 48

Offset: 1

Ilya Gutkovskiy, Sep 07 2019

Moebius transform of A181983.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002129, A008683, A037225, A181983, A319997, A319998.

[&+[MoebiusMu(Floor(n/d))*(-1)^(d+1)*d:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 07 2019

a[n_] := Sum[MoebiusMu[n/d] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - 4 EulerPhi[n/2]]; Table[a[n], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p - 1)*p^(e - 1); f[2, 1] = -3; f[2, e_] := -2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, moebius(n/d)*(-1)^(d+1)*d); \\ Michel Marcus, Sep 07 2019

G.f.: Sum_{k>=1} mu(k) * x^k / (1 + x^k)^2.
G.f. A(x) satisfies: A(x) = x / (1 + x)^2 - Sum_{k>=2} A(x^k).
a(n) = phi(n) if n odd, phi(n) - 4*phi(n/2) if n even, where phi = A000010.
a(n) = A319997(n) - A319998(n).
Multiplicative with a(2) = -3, a(2^e) = -2^(e-1) for e > 1, and a(p^e) = (p-1)*p^(e-1) for p > 2. - Amiram Eldar, Nov 15 2022

A345082 Number of elements of order n in R/Z X Z/2Z.

Original entry on oeis.org

1, 3, 2, 4, 4, 6, 6, 8, 6, 12, 10, 8, 12, 18, 8, 16, 16, 18, 18, 16, 12, 30, 22, 16, 20, 36, 18, 24, 28, 24, 30, 32, 20, 48, 24, 24, 36, 54, 24, 32, 40, 36, 42, 40, 24, 66, 46, 32, 42, 60, 32, 48, 52, 54, 40, 48, 36, 84, 58, 32, 60, 90, 36, 64, 48, 60, 66, 64

Offset: 1

Michel Marcus, Jul 30 2021

From Peter Bala, Dec 30 2023: (Start)
Denoted phi_2(n) in van der Kamp.
The number of solutions of the congruence x*y == 2 (mod n), 1 <= x, y <= n.
Can be regarded as a generalization of Euler's totient function phi(n) = Sum_{k = 1..n, gcd(k,n) = 1} gcd(k,n) since a(n) = Sum_{k = 1..n, gcd(k,n) divides 2} gcd(k,n). (End)

N. Anghel, Heron triangles with constant area and perimeter, Rev. Roumaine Math. Pures Appl. 65 (2020), 4, 403-422.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT], 2012.

Cf. A000010, A088245.

Cf. A319998, A319997.

with(numtheory):
seq(add(d*phi(n/d), d in divisors(igcd(2, n))), n = 1..70); # Peter Bala, Dec 30 2023

Table[If[OddQ[n],EulerPhi[n],If[Mod[n,4]==0,2EulerPhi[n],2EulerPhi[n]+EulerPhi[n/2]]],{n,68}] (* Stefano Spezia, Jul 30 2021 *)

a(n) = if (n%2, eulerphi(n), if (n%4, 2*eulerphi(n) + eulerphi(n/2), 2*eulerphi(n)));

from sympy import totient as phi
def a(n): return phi(n) if n%2 else 2*phi(n)+phi(n//2) if n%4 else 2*phi(n)
print([a(n) for n in range(1, 69)]) # Michael S. Branicky, Jul 30 2021

a(n) = phi(n) if n is odd; 2*phi(n) if n == 0 (mod 4); 2*phi(n) + phi(n/2) if n == 2 (mod 4).
From Ridouane Oudra, Oct 17 2021: (Start)
a(n) = A000010(n) + A319998(n);
a(n) = 2*A000010(n) - A319997(n);
a(n) = Sum_{j = 1..n} gcd(n,j)*cos(4*Pi*j/n). (End)
From Peter Bala, Dec 30 2023: (Start)
a(n) = Sum_{d divides gcd(2,n)} d*phi(n/d), where phi(n) = A000010(n) denotes Euler's totient function.
Sum_{d divides n} a(d) = 2*n for n even, else equals n (van der Kamp, equation 26).
Dirichlet g.f.: zeta(s-1)*(1 + 2^(1-s))/zeta(s).
The Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + 4*x + x^2)/(1 - x^2)^2. See A022998.
Multiplicative with a(2) = 3, a(2^k) = 2^k for k >= 2 and a(p^k) = p*k - p^(k-1) for odd primes p.
If n divides m then a(n) divides 3*a(m). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Jan 18 2024

cp's OEIS Frontend

A319997 a(n) = Sum_{d|n, d is odd} mu(n/d)*d, where mu(n) is Moebius function A008683.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A323237 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291756(n) for all n, except f(1) = -1 and for odd numbers n > 1, f(n) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A325596 a(n) = Sum_{d|n} mu(n/d) * (-1)^(d + 1) * d.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A345082 Number of elements of order n in R/Z X Z/2Z.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula