A106475 -id:A106475

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A344372

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) { 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)) ); }

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

CROSSREFS

Cf. A106475, A344371, A344373.

Negated bisection of A199084.

Cf. A018804, A332794, A368736 - A368741.

KEYWORD

nonn,mult,easy

AUTHOR

Max Alekseyev, May 16 2021

EXTENSIONS

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

STATUS

approved

A199084

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The alternating sum analog of A018804.

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

LINKS

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.

FORMULA

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

MAPLE

A199084 := proc(n)

add((-1)^(k-1)* igcd(k, n), k=1..n) ;

end proc:

seq(A199084(n), n=1..80) ;

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

sign,easy

AUTHOR

R. J. Mathar, Nov 02 2011

STATUS

approved

A344371

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Max Alekseyev, May 16 2021

EXTENSIONS

More terms from Felix Fröhlich, May 19 2021

STATUS

approved

A344373

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

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.

LINKS

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.

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

sign,easy,look

AUTHOR

Max Alekseyev, May 16 2021

EXTENSIONS

More terms from Antti Karttunen, May 16 2021

STATUS

approved

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