A344371 -id:A344371 - 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

A199084 a(n) = Sum_{k=1..n} (-1)^(k+1) 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

Offset: 1

R. J. Mathar, Nov 02 2011

The alternating sum analog of A018804.

a(2n) <= -(2n-1) (cf. A344372). - Max Alekseyev, May 16 2021

Vincenzo Librandi and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Laszlo Toth, Weighted Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.7.7.

Cf. A001620, A018804, A106475, A306016, A344371, A344372, A344373.

A199084 := proc(n)
        add((-1)^(k-1)* igcd(k,n),k=1..n) ;
end proc:
seq(A199084(n),n=1..80) ;

altGCDSum[n_] := Sum[(-1)^(i + 1)GCD[i, n], {i, n}]; Table[altGCDSum[n], {n, 50}] (* Alonso del Arte, Nov 02 2011 *)
Total/@Table[(-1)^(k+1) GCD[k,n],{n,60},{k,n}] (* Harvey P. Dale, May 29 2013 *)

a(n) = sum(k=1, n, (-1)^(k+1)*gcd(k,n)); \\ Michel Marcus, Jun 28 2023

a(2n+1) = 2n+1. - Seiichi Manyama, Dec 09 2016
a(n) = (-1)^(n+1)*A344371(n) = A344373(n) - (-1)^n*n. - Max Alekseyev, May 16 2021
a(2n) = -A344372(n). - Max Alekseyev, May 16 2021
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

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

A368624 a(n) = Sum_{k = 1..n} (-1)^(n+k) * gcd(2*k, n).

Original entry on oeis.org

1, 0, 3, 4, 5, 0, 7, 16, 9, 0, 11, 20, 13, 0, 15, 48, 17, 0, 19, 36, 21, 0, 23, 80, 25, 0, 27, 52, 29, 0, 31, 128, 33, 0, 35, 84, 37, 0, 39, 144, 41, 0, 43, 84, 45, 0, 47, 240, 49, 0, 51, 100, 53, 0, 55, 208, 57, 0, 59, 180, 61, 0, 63, 320, 65, 0, 67, 132, 69, 0

Offset: 1

Peter Bala, Jan 01 2024

Cf. A344371, A344372.

seq(add((-1)^(n+k) * gcd(2*k, n), k = 1..n), n = 1..70)

Table[Sum[(-1)^(n+k) GCD[2k,n],{k,n}],{n,70}] (* Harvey P. Dale, Jun 16 2024 *)

a(2*n+1) = 2*n + 1; a(4*n+2) = 0; a(4*n) = 4*A344372(n) = 4*Sum_{k = 1..n} gcd(2*k, n).

cp's OEIS Frontend

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

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A199084 a(n) = Sum_{k=1..n} (-1)^(k+1) 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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A368624 a(n) = Sum_{k = 1..n} (-1)^(n+k) * gcd(2*k, n).

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