A069097 -id:A069097 - cp's OEIS frontend

A360428 Inverse Mobius transformation of A338164.

1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833, 1024, 577, 1575, 721, 1960, 1649, 1687, 1057, 3536, 1825, 2359, 2673, 3880, 1681, 5831, 1921, 4864, 4097, 4039, 4753, 9000, 2737, 5047, 5729, 10192, 3361, 11543, 3697, 9640, 11025, 7399, 4417, 17408, 7105, 12775

Offset: 1

R. J. Mathar, Feb 07 2023

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

Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.

A360428 := proc(n)
    add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.
Dirichlet convolution of A034714 and A008683.
Dirichlet g.f.: zeta^2(s-2)/zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.
a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

Original entry on oeis.org

1, 10, 33, 88, 145, 330, 385, 736, 945, 1450, 1441, 2904, 2353, 3850, 4785, 6016, 5185, 9450, 7201, 12760, 12705, 14410, 12673, 24288, 18625, 23530, 26001, 33880, 25201, 47850, 30721, 48640, 47553, 51850, 55825, 83160, 51985, 72010, 77649, 106720

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005353, A059306, A062801, A069097, A102631, A240547.

f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2020 *)

a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])

a(n)=local(c=0); forvec(x=vector(4,k,[1,n]),c+=((x[1]*x[2]-x[3]*x[4])%n==0)); c

From Vladeta Jovovic, Apr 22 2002: (Start)
a(n) = n^4 - A005353(n).
Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)
Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).
A102631(n) | a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
From Piotr Rysinski, Sep 11 2020: (Start)
a(n) = n * A069097(n).
Proof: a(n) is multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1), A069097(n) is multiplicative with A069097(p^e) = p^(e-1)*(p^e*(p+1)-1), so a(p^e) = p^e*A069097(p^e). (End)

A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

Original entry on oeis.org

1, 8, 33, 32, 145, 264, 385, 128, 945, 1160, 1441, 1056, 2353, 3080, 4785, 512, 5185, 7560, 7201, 4640, 12705, 11528, 12673, 4224, 18625, 18824, 26001, 12320, 25201, 38280, 30721, 2048, 47553, 41480, 55825, 30240, 51985, 57608, 77649, 18560, 70561, 101640

Offset: 1

Laszlo Toth, Apr 07 2014

			For n=2 the a(2)=8 solutions are (0,0,0,0), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1).

David A. Corneth, Table of n, a(n) for n = 1..10000
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Index to sequences related to sums of squares.

Cf. A020478, A069097, A086933, A087687, A208895, A229179.

A240547 := proc(n) local a, x, y, z, t ; a := 0 ; for x from 0 to n-1 do for y
from 0 to n-1 do for z from 0 to n-1 do for t from 0 to n-1 do if
(x^2+y^2+z^2+t^2) mod n = 0 mod n then a := a+1 ; fi; od; od ; od; od;
a ; end proc;
# alternative
A240547 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1) ;
        else
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A240547(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

b[2, e_] := 2^(2 e + 1);
b[p_, e_] := p^(2 e - 1)*(p^(e + 1) + p^e - 1);
a[n_] := Times @@ b @@@ FactorInteger[n];
Array[a, 42] (* Jean-François Alcover, Dec 05 2017 *)

a(n) = my(m); if( n<1, 0, forvec( v = vector(4, i, [0, n-1]), m += (0 == norml2(v)%n))); m /* Michael Somos, Apr 07 2014 */

a(n) = {my(f = factor(n), res = 1, start = 1, p, e, i); if(n % 2 == 0, res = 1<<(f[1,2]<<1+1); start = 2); for(i = start, #f~, p = f[i, 1]; e = f[i, 2]; res*=(p^(e<<1-1)*(p^(e+1)+p^e-1))); res} \\ David A. Corneth, Jul 22 2018

Multiplicative, with a(2^e) = 2^(2e+1) for e>=1, a(p^e) = p^(2e-1)*(p^(e+1)+p^e-1) for p > 2, e>=1.
For odd n, a(n) = A069097(n)*n = A020478(n). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3 * log(n)), where c = 5*Pi^2/(168*zeta(3)) = 0.244362... (Tóth, 2014). - Amiram Eldar, Oct 18 2022

A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 4, 8, 15, 19, 35, 34, 56, 63, 86, 76, 141, 103, 157, 182, 212, 169, 294, 208, 355, 335, 359, 298, 556, 405, 490, 522, 657, 463, 865, 526, 816, 773, 812, 856, 1239, 739, 1003, 1058, 1424, 901, 1610, 988, 1525, 1617, 1445, 1174, 2188, 1435, 1960, 1760, 2091, 1483, 2529, 1994

Offset: 1

Ilya Gutkovskiy, Jul 23 2019

nonn

Dirichlet convolution of Euler totient function with triangular numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000010, A000217, A018804, A069097, A272718, A309323.

nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1)/2, {d, Divisors[n]}], {n, 1, 55}]
Table[Sum[Sum[GCD[j, k, n], {j, 1, k}], {k, 1, n}], {n, 1, 55}]

a(n) = Sum_{d|n} phi(n/d) * d * (d + 1)/2.
a(n) = Sum_{k=1..n} Sum_{j=1..k} gcd(j,k,n).
a(n) = Sum_{k=1..n} gcd(n,k)*(gcd(n,k)+1)/2. - Richard L. Ollerton, May 07 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (36*zeta(3)). - Vaclav Kotesovec, May 23 2021
a(n) = (A018804(n) + A069097(n))/2. - Ridouane Oudra, May 22 2025

A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

Original entry on oeis.org

1, 19, 89, 316, 649, 1691, 2449, 5104, 7281, 12331, 14761, 28124, 28729, 46531, 57761, 81856, 83809, 138339, 130681, 205084, 217961, 280459, 280369, 454256, 406225, 545851, 590409, 773884, 708121, 1097459, 924481, 1310464, 1313729, 1592371, 1589401, 2300796

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372927.
Cf. A343498, A372929, A372931.
Cf. A001157, A008683.
Cf. A002117, A013663.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 2));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3)/zeta(5) = 1.1592484598... . (End)

A294403 E.g.f.: exp(-Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, -1, -5, -7, 1, 839, 4171, 54305, 102817, -4303441, -74521349, -1595325271, -20768141855, -222701825737, 1485790534411, 65580347824529, 2880129557707201, 67631429234674655, 1543424936566399867, 23542870556917468889, 119940955037901088321

Offset: 0

Seiichi Manyama, Oct 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..439

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): A294402 (k=0), this sequence (k=1), A294404 (k=2).

Cf. A000203, A010815, A069097, A294361.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sigma(k)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j)^2. - Ilya Gutkovskiy, Aug 17 2021

A332654 a(n) = Sum_{k=1..n} (k/gcd(n, k))^2.

Original entry on oeis.org

1, 2, 6, 12, 31, 33, 92, 96, 165, 172, 386, 239, 651, 499, 656, 776, 1497, 846, 2110, 1262, 1903, 2037, 3796, 1867, 4181, 3408, 4530, 3673, 7715, 3183, 9456, 6232, 7761, 7754, 10062, 6248, 16207, 10889, 12980, 9906, 22141, 9308, 25586, 15027, 17075, 19483, 33512, 14851

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Inverse Moebius transform of A053818.

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

Cf. A053818, A057661, A068963, A069097, A320941, A332655.

[&+[(k div Gcd(n,k))^2:k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[(k/GCD[n, k])^2, {k, 1, n}], {n, 1, 48}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^2, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 48}]

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^2.

a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} k^2.

A321294 a(n) = Sum_{d|n} mu(n/d)dsigma_n(d).

Original entry on oeis.org

1, 9, 83, 1058, 15629, 282381, 5764807, 134480900, 3486902505, 100048836321, 3138428376731, 107006403495850, 3937376385699301, 155572843119518781, 6568408661060858767, 295150157013526773768, 14063084452067724991025, 708236697425777157039381

Offset: 1

Ilya Gutkovskiy, Nov 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A018804, A069097, A320940, A332517, A342432, A342433.

Table[Sum[MoebiusMu[n/d] d DivisorSigma[n, d], {d, Divisors[n]}], {n, 18}]
Table[Sum[EulerPhi[n/d] d^(n + 1), {d, Divisors[n]}], {n, 18}]
Table[Sum[GCD[n, k]^(n + 1), {k, n}], {n, 18}]

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n)); \\ Michel Marcus, Nov 03 2018

from sympy import totient, divisors
def A321294(n):
    return sum(totient(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = [x^n] Sum_{i>=1} Sum_{j>=1} mu(i)*j^(n+1)*x^(i*j)/(1 - x^(i*j))^2.
a(n) = Sum_{d|n} phi(n/d)*d^(n+1).
a(n) = Sum_{k=1..n} gcd(n,k)^(n+1).
a(n) ~ n^(n+1). - Vaclav Kotesovec, Nov 02 2018

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Original entry on oeis.org

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

Table[Sum[DivisorSigma[1, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[1, n^2] + 2*Sum[DivisorSigma[1, j*n], {j, 1, n - 1}], {n, 2, 50}]]

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

A033457 GCD-convolution of squares A000290 with themselves.

Original entry on oeis.org

1, 2, 6, 4, 19, 6, 28, 24, 45, 10, 98, 12, 79, 94, 120, 16, 201, 18, 238, 164, 171, 22, 436, 120, 229, 234, 426, 28, 695, 30, 496, 352, 369, 370, 1014, 36, 451, 470, 1068, 40, 1261, 42, 946, 1020, 639, 46, 1832, 336, 1225, 754, 1278, 52, 1899, 774, 1924, 920, 981

Offset: 0

N. J. A. Sloane

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A000010 (phi), A000290, A069097, A306633.

Table[Sum[d^2*EulerPhi[(n + 2)/d], {d, Most@ Divisors[n + 2]}], {n, 0, 47}] (* Michael De Vlieger, Mar 20 2015 *)
f[p_, e_] := p^(e - 1)*(p^e*(p + 1) - 1); a[n_] := Times @@ f @@@ FactorInteger[n + 2] - (n + 2)^2; Array[a, 100, 0] (* Amiram Eldar, Dec 06 2024 *)

a(n) = {my(f = factor(n+2)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(e-1)*(p^e*(p+1) - 1)) - (n+2)^2;} \\ Amiram Eldar, Dec 06 2024

sum([d^2*euler_phi(int((n+2)/d)) for d in range(1,n+2) if (n+2)%d==0]) # Danny Rorabaugh, Mar 20 2015

a(n-2) = Sum_{d|n, dVladeta Jovovic, Aug 27 2003
From Amiram Eldar, Dec 06 2024: (Start)
a(n) = A069097(n+2) - (n+2)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/zeta(3) - 1)/3 = (A306633 - 1)/3 = 0.122810925... . (End)

cp's OEIS Frontend

A360428 Inverse Mobius transformation of A338164.

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A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

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A240547 Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

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A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

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A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

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A294403 E.g.f.: exp(-Sum_{n>=1} sigma(n) * x^n).

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A332654 a(n) = Sum_{k=1..n} (k/gcd(n, k))^2.

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A321294 a(n) = Sum_{d|n} mu(n/d)*d*sigma_n(d).

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A321294 a(n) = Sum_{d|n} mu(n/d)dsigma_n(d).