A199084 -id:A199084 - cp's OEIS frontend

A344372 a(n) = Sum_{k = 1..n} gcd(2*k, n).

1, 4, 5, 12, 9, 20, 13, 32, 21, 36, 21, 60, 25, 52, 45, 80, 33, 84, 37, 108, 65, 84, 45, 160, 65, 100, 81, 156, 57, 180, 61, 192, 105, 132, 117, 252, 73, 148, 125, 288, 81, 260, 85, 252, 189, 180, 93, 400, 133, 260, 165, 300, 105, 324, 189, 416, 185, 228, 117, 540, 121, 244, 273

Offset: 1

Max Alekseyev, May 16 2021

For all n, a(n) >= 2*n - 1, where the equality holds if n is 1 or an odd prime.

a(n) equals the number of solutions to the congruence 2*x*y == 0 (mod n) for 1 <= x, y <= n. - Peter Bala, Jan 11 2024

			a(6) = 20: the 20 solutions to the congruence 2*x*y == 0 (mod 6) for 1 <= x, y <= 6 are the pairs (x, y) = (k, 6) for 1 <= k <= 6, the pairs (6, k) for 1 <= k <= 5, the pairs (3, k) for 1 <= k <= 5 and the pairs (1, 3), (2, 3), (4, 3) and (5, 3). - _Peter Bala_, Jan 11 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A106475, A344371, A344373.
Negated bisection of A199084.
Cf. A018804, A332794, A368736 - A368741.

seq(add((-1)^k*gcd(k, 2*n), k = 1..2*n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(gcd(2,d)*phi(d)*n/d, d in divisors(n)), n = 1..70); # Peter Bala, Jan 08 2024

f[p_, e_] := (e + 1)*p^e - e*p^(e - 1); f[2, e_] := (e + 1)*2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 20 2022 *)
Table[Sum[GCD[2*k, n], {k, 1, n}], {n, 1, 60}] (* or *)
Table[Sum[(-1)^k * GCD[k, 2*n], {k, 1, 2*n}], {n, 1, 60}] (* Vaclav Kotesovec, Jan 13 2024 *)

{ A344372(n) = my(f=factor(n)); prod(i=1,#f~, (f[i,2]+1)*f[i,1]^f[i,2] - if(f[i,1]>2,f[i,2]*f[i, 1]^(f[i,2]-1)) ); }

a(n) = sum(k=1, 2*n, (-1)^k*gcd(k,2*n)); \\ Michel Marcus, May 17 2021

a(n) = Sum_{k = 1..2*n} (-1)^k * gcd(k,2*n).
a(n) is multiplicative with a(2^d) = (d+1)*2^d, and a(p^d) = (d+1)*p^d - d*p^(d-1) for an odd prime p, d >= 1.
a(n) = A344371(2*n) = -A199084(2*n) = 2*n - A106475(n-1).
a(n) = A018804(n) if n is odd, 4*A018804(n/2) if n is even. - Sebastian Karlsson, Aug 31 2021
From Peter Bala, Jan 11 2023: (Start)
a(n) = Sum_{d divides n} phi(2*d)*n/d, where phi(n) = A000010(n).
a(n) = - A332794(2*n); a(2*n+1) = A368736(2*n+1).
Dirichlet g.f.: 1/(1 - 1/2^s) * zeta(s-1)^2/zeta(s).
Define D(n) = Sum_{d divides n} a(d). Then
D(2*n+1) = (2*n + 1)*tau(2*n+1), where tau(n) = A000005(n), the number of divisors of n.
The sequence {(1/4)*( D(2*n) - D(n) ) : n >= 1} begins {1, 3, 6, 8, 10, 18, 14, 20, 27, 30, 22, 48, 26, 42, 60, 48, 34, 81, 38, 80, 84, 66, ...} and appears to be multiplicative. (End)
Sum_{k=1..n} a(k) ~ 4*n^2 * (log(n) - 1/2 + 2*gamma - log(2)/3 - 6*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 12 2024

New name according to the formula by Peter Bala from Vaclav Kotesovec, Jan 13 2024

A344371 a(n) = Sum_{k=1..n} (-1)^(n-k) gcd(k,n).

Original entry on oeis.org

1, 1, 3, 4, 5, 5, 7, 12, 9, 9, 11, 20, 13, 13, 15, 32, 17, 21, 19, 36, 21, 21, 23, 60, 25, 25, 27, 52, 29, 45, 31, 80, 33, 33, 35, 84, 37, 37, 39, 108, 41, 65, 43, 84, 45, 45, 47, 160, 49, 65, 51, 100, 53, 81, 55, 156, 57, 57, 59, 180, 61, 61, 63, 192, 65, 105

Offset: 1

Max Alekseyev, May 16 2021

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A001620, A090423, A106475, A199084, A306016, A344372, A344373.

a(n) = sum(k=1, n, (-1)^(n-k)*gcd(k,n)); \\ Michel Marcus, May 16 2021

a(n) = abs(A199084(n)).
a(2n+1) = 2n+1.
a(2n) = A344372(n) = 2*n - A106475(n-1).
Sum_{k=1..n} a(k) ~ (n^2/Pi^2) * (log(n) + 2*gamma - 1/2 - 4*log(2)/3 + Pi^2/4 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

More terms from Felix Fröhlich, May 19 2021

A106475 An alternating sum of greatest common divisors.

Original entry on oeis.org

1, 0, 1, -4, 1, -8, 1, -16, -3, -16, 1, -36, 1, -24, -15, -48, 1, -48, 1, -68, -23, -40, 1, -112, -15, -48, -27, -100, 1, -120, 1, -128, -39, -64, -47, -180, 1, -72, -47, -208, 1, -176, 1, -164, -99, -88, 1, -304, -35, -160, -63, -196, 1, -216, -79, -304, -71, -112, 1, -420, 1, -120, -147, -320, -95, -288, 1, -260, -87

Offset: 0

Paul Barry, May 03 2005

With interpolated 0's, this is Sum_{k=0..n} gcd(n-k+1,k+1)*(-1)^k.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A001620, A003989, A006579, A199084, A306016, A344371, A344372.

Negated bisection of A344373.

Table[Sum[GCD[2n-k+1,k+1](-1)^k,{k,0,2n}],{n,0,100}] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

A106475(n) = sum(k=0,(2*n),gcd(1+n+n-k, k+1)*((-1)^k)); \\ Antti Karttunen, Mar 30 2021

a(n) = Sum_{k=0..2*n} gcd(2*n-k+1, k+1)*(-1)^k.
a(n) = 2(n+1) - A344371(2(n+1)) = 2(n+1) - A344372(n+1) = 2(n+1) + A199084(2(n+1)). - Max Alekseyev, May 16 2021
Sum_{k=1..n} a(k) ~ n^2 * (1 - (4/Pi^2)*(log(n) + 2*gamma - 1/2 - log(2)/3 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

More terms from Antti Karttunen, Mar 30 2021

A344373 a(n) = Sum_{k=1..n-1} (-1)^k gcd(k, n).

Original entry on oeis.org

0, -1, 0, 0, 0, -1, 0, 4, 0, -1, 0, 8, 0, -1, 0, 16, 0, 3, 0, 16, 0, -1, 0, 36, 0, -1, 0, 24, 0, 15, 0, 48, 0, -1, 0, 48, 0, -1, 0, 68, 0, 23, 0, 40, 0, -1, 0, 112, 0, 15, 0, 48, 0, 27, 0, 100, 0, -1, 0, 120, 0, -1, 0, 128, 0, 39, 0, 64, 0, 47, 0, 180, 0, -1, 0, 72, 0, 47, 0, 208, 0, -1, 0, 176, 0, -1, 0, 164, 0, 99

Offset: 1

Max Alekseyev, May 16 2021

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

For all n, a(n) >= -1. Equality holds for n = 2 and n = 2*p for an odd prime p.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Pruthviraj et al., Is it always true that Sum_{i=1..a-1}(-1)^i(a,i) >= -1 ?, 2021.

Cf. A001620, A106475, A199084, A306016, A344371, A344372.

Array[Sum[(-1)^k GCD[k, #], {k, # - 1}] &, 90] (* Michael De Vlieger, May 16 2021 *)

A344373(n) = sum(k=1,n-1,((-1)^k)*gcd(k,n)); \\ Antti Karttunen, May 16 2021

a(n) = -A199084(n) - (-1)^n*n = (-1)^n * (A344371(n) - n).
a(2*n+1) = 0.
a(2*n) = A344372(n) - 2*n = -A106475(n-1).
Sum_{k=1..n} a(k) ~ (n^2/4) * ((4/Pi^2)*(log(n) + 2*gamma - 1/2 - 4*log(2)/3 - zeta'(2)/zeta(2)) - 1), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

More terms from Antti Karttunen, May 16 2021

A332794 a(n) = Sum_{d|n} (-1)^(d + 1) * d * phi(n/d).

Original entry on oeis.org

1, -1, 5, -4, 9, -5, 13, -12, 21, -9, 21, -20, 25, -13, 45, -32, 33, -21, 37, -36, 65, -21, 45, -60, 65, -25, 81, -52, 57, -45, 61, -80, 105, -33, 117, -84, 73, -37, 125, -108, 81, -65, 85, -84, 189, -45, 93, -160, 133, -65, 165, -100, 105, -81, 189

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.

Cf. A000010, A018804, A078747, A143520, A181983, A193356, A199084, A344372, A368929.

a[n_] := Sum[(-1)^(d + 1) d EulerPhi[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[OddQ[n], Sum[GCD[n, k], {k, 1, n}], Sum[(-1)^(k + 1) GCD[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e*(p-1) + p)*p^(e-1); f[2, e_] := -e*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)

a(n) = sumdiv(n, d, (-1)^(d+1)*d*eulerphi(n/d)); \\ Michel Marcus, Feb 24 2020

G.f.: Sum_{k>=1} phi(k) * x^k / (1 + x^k)^2.
Dirichlet g.f.: zeta(s-1)^2 * (1 - 2^(2 - s)) / zeta(s).
a(n) = Sum_{k=1..n} gcd(n, k) if n odd, Sum_{k=1..n} (-1)^(k + 1) * gcd(n, k) if n even.
From Amiram Eldar, Nov 04 2022: (Start)
Multiplicative with a(2^e) = -e*2^(e-1), and a(p^e) = (e*(p-1) + p)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*log(2)/Pi^2 = 0.210691... . (End)
a(2*n) = - Sum_{k = 1..n} gcd(2*k, n) = - A344372(n); a(2*n+1) = A018804(2*n+1). - Peter Bala, Jan 11 2024
a(n) = Sum_{k = 1..n} (-1)^(1 + gcd(k, n)) * gcd(k, n) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

A199806 Alternating LCM-sum: a(n) = Sum_{k=1..n} (-1)^(k-1)*lcm(k,n).

Original entry on oeis.org

1, 0, 0, 8, -5, 18, -14, 80, -9, 100, -44, 204, -65, 294, 30, 672, -119, 540, -152, 1040, 63, 1210, -230, 1752, -75, 2028, -54, 2996, -377, 2190, -434, 5440, 165, 4624, 280, 5472, -629, 6498, 234, 8800, -779, 6300, -860, 12188, 225, 11638, -1034, 14256, -245, 13000

Offset: 1

Laszlo Toth, Nov 10 2011

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7

Cf. A051193, A199084.

[&+[(-1)^(k-1)*Lcm(k,n):k in [1..n]]: n in [1..50]]; // Marius A. Burtea, Oct 02 2019

Table[Sum[(-1)^(k-1) LCM[k,n],{k,n}],{n,50}] (* Harvey P. Dale, Jan 30 2024 *)

a(n)=-sum(k=1,n,(-1)^k*lcm(k,n)) \\ Charles R Greathouse IV, Nov 10 2011

def A199806(n) : return add((-1)^(k-1)*lcm(k,n) for k in (1..n))
[A199806(n) for n in (1..50)]  # Peter Luschny, Nov 10 2011

A333493 a(n) = Sum_{k=1..n} (-1)^(k+1) * lcm(n,k) / gcd(n,k).

Original entry on oeis.org

1, 1, -2, 13, -9, 28, -20, 109, -11, 151, -54, 256, -77, 442, 48, 877, -135, 757, -170, 1363, 103, 1816, -252, 2080, -59, 3043, -38, 3982, -405, 2878, -464, 7021, 273, 6937, 390, 6817, -665, 9748, 388, 11059, -819, 8407, -902, 16348, 219, 17458, -1080, 16672, -167

Offset: 1

Ilya Gutkovskiy, Mar 24 2020

sign

Alternating row sums of A051537.

Cf. A056789, A089913, A199084, A199806.

Table[Sum[(-1)^(k + 1) LCM[n, k]/GCD[n, k], {k, 1, n}], {n, 1, 49}]

a(n) = sum(k=1, n, (-1)^(k+1)*lcm(n, k)/gcd(n, k)); \\ Michel Marcus, Mar 24 2020

If n odd, a(n) = (1/2) * n * Sum_{d|n} Sum_{j|d} (-1)^(j + 1) * mu(d/j) * (n + d) / j^2.

If n even, a(n) = (1/2) * n^2 * Sum_{d|n} Sum_{j|d} (-1)^(j + 1) * mu(d/j) * (n + d) / (d * j^2).

cp's OEIS Frontend

A344372 a(n) = Sum_{k = 1..n} gcd(2*k, n).

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A344371 a(n) = Sum_{k=1..n} (-1)^(n-k) gcd(k,n).

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A106475 An alternating sum of greatest common divisors.

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A344373 a(n) = Sum_{k=1..n-1} (-1)^k gcd(k, n).

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A332794 a(n) = Sum_{d|n} (-1)^(d + 1) * d * phi(n/d).

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A199806 Alternating LCM-sum: a(n) = Sum_{k=1..n} (-1)^(k-1)*lcm(k,n).

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A333493 a(n) = Sum_{k=1..n} (-1)^(k+1) * lcm(n,k) / gcd(n,k).

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