A306016 -id:A306016 - cp's OEIS frontend

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

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

A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 2, 2, 4, 4, -1, 2, 2, 2, 2, 4, 4, 2, 0, 1, 4, 0, 2, 2, 8, 2, -2, 4, 4, 4, -1, 2, 4, 4, 0, 2, 8, 2, 2, 2, 4, 2, -2, 1, 2, 4, 2, 2, 0, 4, 0, 4, 4, 2, 4, 2, 4, 2, -3, 4, 8, 2, 2, 4, 8, 2, -4, 2, 4, 2, 2, 4, 8, 2, -2, -1, 4, 2, 4, 4, 4, 4, 0, 2, 4, 4, 2, 4, 4, 4, -4, 2, 2, 2

Offset: 1

Labos Elemer

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A008683, A034444, A048105.
Cf. A048107, A048108, A048109, A048111.
Cf. A001620, A306016.

Table[2^(1 + PrimeNu@ n) - DivisorSigma[0, n], {n, 99}] (* Michael De Vlieger, Aug 01 2017 *)

A048106(n) = (2^(1+omega(n)) - numdiv(n)); \\ Antti Karttunen, May 25 2017

from sympy import divisor_count, primefactors
def a(n): return 1 if n==1 else 2**(1 + len(primefactors(n))) - divisor_count(n) # Indranil Ghosh, May 25 2017

a(n) = 2^(1+omega(n)) - d(n) = 2^(1+A001221(n)) - A000005(n).
a(n) = -Sum_{ d divides n } (-1)^mu(d). - Vladeta Jovovic, Jan 24 2002
From Amiram Eldar, Dec 09 2022: (Start)
a(n) > 0 iff n is in A048107.
a(n) < 0 iff n is in A048111.
a(n) <= 0 iff n is in A048108.
a(n) = 0 iff n is in A048109.
Dirichlet g.f: zeta(s)^2*(2/zeta(2*s) - 1).
Sum_{k=1..n} a(k) ~ (12/Pi^2 - 1)*n*log(n) + ((12/Pi^2-1)*(2*gamma-1) - (24/Pi^2)*zeta'(2)/zeta(2))*n, where gamma is Euler's constant (A001620). (End)

A307000 Number of unitary rings with additive group (Z/nZ)^2. Equivalently, number of unitary commutative rings with additive group (Z/nZ)^2.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 3, 10, 5, 9, 3, 18, 3, 9, 9, 14, 3, 15, 3, 18, 9, 9, 3, 30, 5, 9, 7, 18, 3, 27, 3, 18, 9, 9, 9, 30, 3, 9, 9, 30, 3, 27, 3, 18, 15, 9, 3, 42, 5, 15, 9, 18, 3, 21, 9, 30, 9, 9, 3, 54, 3, 9, 15, 22, 9, 27, 3, 18, 9, 27, 3, 50, 3, 9, 15, 18, 9, 27

Offset: 1

Jianing Song, Mar 24 2019

Equivalently, a(n) is the number of nonisomorphic unitary rings whose rank is 2 when viewed as a free module over the ring (Z_n, +, *). - Jianing Song, Feb 23 2021
Every unitary ring with additive group (Z/nZ)^2 must be commutative, and is of the form Z_n[x]/(x^2 + b*x + c) for some b, c in Z_n, where (x^2 + b*x + c) stands for the ideal of Z_n[x] generated by x^2 + b*x + c. Proof: Let R be a unitary commutative ring with additive group (Z/nZ)^2. Suppose e is the identity element of R, x is an element such that {e, x} is a basis for R as a free module over Z_n (such a basis must exist, see my note in the link section), then every element can be written as the form u*x + v*e for 0 <= u, v <= n-1. If x^2 = -p*x - q*e, it turns out that R is isomorphic to Z_n[x]/(x^2 + [p]*x + [q]). - Jianing Song, Apr 23 2021
Equivalently, a(n) is the number of nonisomorphic rings of the form Z[x]/(n, x^2 + p*x + q), where (n, x^2 + p*x + q) is the ideal of Z[x] generated by n and x^2 + p*x + q. - Jianing Song, Feb 15 2021
Theorem. R_1 = Z_n[x]/(x^2 + b*x + c) and R_2 = Z_n[y]/(y^2 + b'*y + c') are isomorphic if and only if there exists some k in Z, t in Z_n such that gcd(k, n) = 1 and that b' == b*k + 2*t (mod n), c' == t^2 + b*k*t + c*k^2 (mod n).
Proof: "<=": Note that y^2 + (b*k + 2*t)*y + (t^2 + b*k*t + c*k^2) = (y + t)^2 + b*k*(y + t) + c*k^2, so a mapping from R_1 to R_2 is given by f(x) = (y + t)/k and f(r*x + s) = r*f(x) + s. Since gcd(k, n) = 1, f is an isomorphic mapping.
"=>": If R_1 and R_2 are isomorphic, there exists some isomorphic mapping from R_2 to R_1 such that f(y) = k*x - t. If gcd(k, n) > 1, since f(r*y + s) = r*f(y) + s = r*(k*x - t), there is no element in R_2 such that f(y) = x, a contradiction. So this isomorphic mapping sends x in R_1 to (y + t)/k, then (y + t)^2 + b*k*(y + t) + c*k^2 = 0. The corresponding coefficients must be equal modulo n, so b' == b*k + 2*t (mod n), c' == t^2 + b*k*t + c*k^2 (mod n).
Now note that without loss of generality we can suppose that b = 0 or -1, because we can always find some t such that b*k + 2*t == 0 or -1 (mod n). Furthermore, if n is an odd number, we can suppose that b = 0.
Case (i): n is an odd number, then a unitary ring with additive group (Z/nZ)^2 is of the form Z_n[x]/(x^2 - c). From the theorem above we can see that R_1 = Z_n[x]/(x^2 - c) and R_2 = Z_n[y]/(y^2 - c') are isomorphic if and only if there exists some k such that gcd(k, n) = 1 and that c*k^2 == c' (mod n). So the number of such rings is A092089(n).
Case (ii): n is an even number, then a unitary ring with additive group (Z/nZ)^2 is of the form Z_n[x]/(x^2 - c) or Z_n[x]/(x^2 - x - (c - 1)/4), c in Z_{4n}, c == 1 (mod 4). From the theorem above we can see that R_1 = Z_n[x]/(x^2 - c) and R_2 = Z_n[y]/(y^2 - c') are isomorphic if and only if there exists some k such that gcd(k, n) = 1 and that c*k^2 == c' (mod n) or c*k^2 + n^2/4 == c' (mod n) (with t = 0 and t = n/2 respectively); R_3 = Z_n[x]/(x^2 - x - (c - 1)/4)) and R_4 = Z_n[y]/(y^2 - y - (c' - 1)/4)) are isomorphic if and only if there exists some k such that gcd(k, 4*n) = 1 and that c*k^2 == c' (mod 4*n) or c*k^2 - n^2 + 2*n == c' (mod 4*n) (with t = (k - 1)/2 and t = (n + k - 1)/2 respectively).
(a) if n == 2 (mod 4), then the number of rings of the form is Z_n[x]/(x^2 - c) is A092089(n/2), and the number of rings of the form Z_n[x]/(x^2 - x - (c - 1)/4) is equal to the number of inequivalent residue classes modulo 4*n that are congruent to 1 modulo 4 where the equivalence relation is defined as [a] ~ [b] (mod 4*n) if and only if there exists some k such that gcd(k, 4*n) = 1 and that a*k^2 == b (mod 4*n). The number of the even inequivalent residue classes modulo 4*n is equal to the number of inequivalent residue classes modulo 2*n, and the number of inequivalent residue classes modulo 4*n that are congruent to 1 modulo 4 is equal to the number of those that are congruent to 3 modulo 4. So the total number if A092089(n/2) + (A092089(4*n) - A092089(2*n))/2.
(b) if n == 0 (mod 4). Similarly, the number of rings of the form is Z_n[x]/(x^2 - c) is A092089(n), and the number of rings of the form Z_n[x]/(x^2 - x - (c - 1)/4) is (A092089(2*n) - A092089(n))/2.

			The nonisomorphic unitary rings with additive group (Z/nZ)^2 (rings of the form Z_n[x]/(x^2 + b*x + c)) are given by Z_n[x]/(f(x)), where f(x) =
n = 1: x^2 (total number = 1);
n = 2: x^2, x^2 - x, x^2 - x - 1 (total number = 3);
n = 3: x^2, x^2 - 1, x^2 - 2 (total number = 3);
n = 4: x^2, x^2 - 1, x^2 - 2, x^2 - 3, x^2 - x, x^2 - x - 1 (total number = 6);
n = 5: x^2, x^2 - 1, x^2 - 2 (total number = 3);
n = 6: x^2, x^2 - 1, x^2 - 2, x^2 - x, x^2 - x - 1, x^2 - x - 2, x^2 - x - 3, x^2 - x - 4, x^2 - x - 5 (total number = 9);
n = 7: x^2, x^2 - 1, x^2 - 3 (total number = 3);
n = 8: x^2, x^2 - 1, x^2 - 2, x^2 - 3, x^2 - 4, x^2 - 5, x^2 - 6, x^2 - 7, x^2 - x, x^2 - x - 1 (total number = 10);
n = 9: x^2, x^2 - 1, x^2 - 2, x^2 - 3, x^2 - 6 (total number = 5);
n = 10: x^2, x^2 - 1, x^2 - 2, x^2 - x, x^2 - x - 1, x^2 - x - 3, x^2 - x - 4, x^2 - x - 5, x^2 - x - 6 (total number = 9).
See the link for rings of the form Z_n[x]/(x^2 + b*x + c) for n <= 100.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christof Nöbauer, Numbers of rings on groups of prime power order.
Jianing Song, List of rings of the form Z_n[x]/(x^2 + b*x + c) for n <= 100.
Jianing Song, Note on A307000.

Cf. A092089.
Cf. also A341547, A341548, A341201, A341202.
Cf. A001620, A082633, A201994, A306016.

f[2, e_] := If[e == 1, 3, 4*e - 2]; f[p_, e_] := 2*e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p>=3, r*=(2*e+1));
        if(p==2&&e==1, r*=3);
        if(p==2&&e>=2, r*=4*e-2);
    );
    return(r);
}

a(n) = A092089(n) if n is odd; (A092089(n) + A092089(2*n))/2 if n is even.
Multiplicative with a(p^e) = 2*e + 1, a(2) = 3 and a(2^e) = 4*e - 2 for e >= 2.
Dirichlet g.f.: zeta(s)^3/zeta(2s)*(1/(1+2^(-s))).
Sum_{k=1..n} a(k) ~ (2*n/Pi^2) * (log(n)^2 + c_1 * log(n) + c_2), where c_1 = 6 * gamma - 2 + 2*log(2)/3 - 4*zeta'(2)/zeta(2) = 4.2052360821..., gamma is Euler's constant (A001620), c_2 = 2 - 6*gamma + 6*gamma^2 - 2*log(2)/3 + 2*gamma*log(2) - log(2)^2/9 - 6*gamma_1 + 4*(1 - 3*gamma - log(2)/3)*zeta'(2)/zeta(2) + 8*(zeta'(2)/zeta(2))^2 - 4*zeta''(2)/zeta(2) = 1.2136692558..., and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Dec 22 2023

New name from Jianing Song, Feb 15 2021

New name from Jianing Song, Apr 23 2021

A174961 Number of non-unitary divisors of the n-th nonsquarefree number.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 4, 1, 2, 2, 4, 5, 4, 2, 2, 6, 1, 2, 2, 4, 4, 4, 2, 5, 2, 8, 2, 2, 6, 3, 4, 4, 4, 2, 8, 2, 2, 5, 4, 8, 6, 2, 2, 8, 1, 2, 2, 4, 6, 4, 4, 4, 4, 11, 2, 2, 4, 4, 2, 4, 8, 6, 2, 8, 1, 2, 2, 2, 6, 10, 4, 2, 4, 10, 5, 4, 8, 4, 2, 6, 2, 12, 4, 8, 5, 4, 4, 4, 2, 12, 2, 4, 2

Offset: 1

N. Wu (neil_wu0626(AT)yahoo.com), Apr 02 2010

nonn

The nonzero terms of A048105.

Also number of nonsquarefree divisors of the n-th nonsquarefree number. The terms in A013929 which correspond to records in this sequence are given by A309141(n); n >= 2. - David James Sycamore, Jan 07 2025

			For n = 4, the fourth nonsquarefree number is A013929(4) = 12 which has 2 non-unitary divisors, 2 and 6. Therefore a(4) = 2.
The number of nonsquarefree divisors of 12 is also = 2 (4 and 12). For n = 55, A013929(55) = 144 so by the third formula above a(55) = A000005(144) - A000005(6) = 15 - 4 = 11 = number of nonsquarefree divisors of 144 (4,8,9,12,16,18,24,36,48,72,144). - _David James Sycamore_, Jan 07 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Cf. A013929, A048105.
Cf. A001620, A013661, A306016.
Cf. A034444, A309141.

Select[Table[DivisorSigma[0, n] - 2^(PrimeNu[n]), {n, 1, 500}], # > 0 &] (* G. C. Greubel, May 21 2017 *)

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(numdiv(f) - 2^omega(f), ", ")));} \\ Amiram Eldar, Dec 09 2023

from math import prod, isqrt
from sympy import mobius, factorint
def A174961(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return -(1<Chai Wah Wu, Aug 12 2024

From Amiram Eldar, Dec 09 2023: (Start)
a(n) = A048105(A013929(n)).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
a(n) = A000005(A013929(n)) - A000005(A007947(A013929(n))). - David James Sycamore, Jan 07 2025

Edited by Amiram Eldar, Dec 09 2023

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

A370895 Partial alternating sums of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, -2, 3, -5, 4, -11, 2, -18, 3, -24, -3, -43, -18, -57, -12, -60, -27, -90, -53, -125, -60, -123, -78, -178, -113, -188, -107, -211, -154, -289, -228, -340, -235, -334, -217, -385, -312, -423, -298, -478, -397, -592, -507, -675, -486, -621, -528, -768, -635, -830

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A018804, A272718.
Cf. A001620, A306016.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(#+1) * pil[#] &, 100]]

pil(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2]);}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pil(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A018804(k).

a(n) = -(1/Pi^2) * n^2 * (log(n) + 2*gamma - 1/2 - zeta'(2)/zeta(2) - 10*log(2)/3) + O(n^(547/416 + eps)), where gamma is Euler's constant (A001620) (Tóth, 2017).

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

A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 142, 255, 59, 77, 63, 321, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 374, 235, 95, 301, 162, 43, 245, 459, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315, 170, 769, 297

Offset: 1

Ilya Gutkovskiy, Aug 27 2018

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).

Cf. A001620, A013661, A306016.

f:= proc(n) local d;
   numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
map(f, [$1..65]); # Robert Israel, Jan 13 2025

Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

a(n) = numerator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Numerators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Numerators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = numerator of Sum_{d|n} sigma(d)/d.
a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).
If p is a prime, a(p) = 2*p + 1.
Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

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

A381456 Decimal expansion of Product_{p prime} p^(1/(p^2-1)).

Original entry on oeis.org

1, 7, 6, 8, 1, 9, 8, 0, 7, 8, 1, 5, 3, 2, 4, 4, 9, 8, 4, 1, 3, 0, 8, 5, 3, 0, 7, 7, 2, 3, 1, 4, 9, 6, 5, 5, 2, 3, 1, 2, 9, 4, 2, 2, 8, 5, 9, 1, 2, 5, 8, 9, 7, 6, 1, 2, 5, 3, 0, 1, 4, 1, 3, 7, 5, 8, 6, 1, 0, 7, 9, 1, 4, 6, 0, 0, 0, 0, 4, 3, 0, 0, 9, 3, 0, 3, 1, 5, 7, 1, 7, 1, 0, 7, 2, 8, 5, 1, 5, 6, 1, 9, 3, 8, 0, 6, 6, 6

Offset: 1

Jwalin Bhatt, Feb 24 2025

The geometric mean of the zeta distribution with parameter value 2 (A381522) approaches this constant.

In general, for parameter value `s` it approaches e^(-zeta'(s)/zeta(s)). - Jwalin Bhatt, Feb 26 2025

			1.768198078153244984130853077...

Cf. A000040, A001620, A074962, A013661, A085548, A306016, A381522, A381898.

Mathematica
```
N[Exp[-Zeta'[2]/Zeta[2]], 120]
```

exp(-zeta'(2)/zeta(2)) \\ Amiram Eldar, Feb 24 2025

from mpmath import zeta, diff, exp, mp
mp.dps = 120
const = exp(-diff(zeta, 2)/zeta(2))
A381456 = [int(d) for d in mp.nstr(const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Apr 08 2025

N(exp(-diff(zeta(s:=var('s')), s).subs(s==2) / zeta(2)), 120)

Equals Product_{p>=2} p^(1/(p^2-1)) where p is prime.
Equals (A^12)/(2*Pi*(e^gamma)) where A = A074962 is the Glaisher-Kinkelin constant and gamma = A001620 is the Euler-Mascheroni constant.
Equals e^(-zeta'(2)/zeta(2)).
Equals exp((Sum_{k>=2} log(k)/(k^2))*(6/(Pi^2))).
Equals (Product_{k>=2} k^(1/(k^2)))^(6/(Pi^2)).
Equals exp(A306016). - Hugo Pfoertner, Feb 24 2025

cp's OEIS Frontend

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

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A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

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A307000 Number of unitary rings with additive group (Z/nZ)^2. Equivalently, number of unitary commutative rings with additive group (Z/nZ)^2.

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A174961 Number of non-unitary divisors of the n-th nonsquarefree number.

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

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A370895 Partial alternating sums of Pillai's arithmetical function (A018804).

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A106475 An alternating sum of greatest common divisors.

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