A068963 -id:A068963 - cp's OEIS frontend

A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

1, 1, 3, 9, 20, 52, 105, 253, 536, 1142, 2421, 4999, 10278, 20686, 41512, 81984, 161029, 312681, 603070, 1153284, 2189331, 4129537, 7733317, 14399693, 26644337, 49034811, 89741600, 163411148, 296074694, 533909026, 958416113, 1712893825, 3048468607, 5403248469, 9539609984

Offset: 0

Paul D. Hanna, May 26 2013

nonn

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A068963, A002618, A061255, A299069, A301986.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m,d,eulerphi(d^3))*x^m/m)+x*O(x^n)),n)}
for(n=0,35,print1(a(n),", "))

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

A069097 Moebius transform of A064987, n*sigma(n).

Original entry on oeis.org

1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319, 376, 305, 525, 379, 638, 605, 655, 551, 1012, 745, 905, 963, 1210, 869, 1595, 991, 1520, 1441, 1525, 1595, 2310, 1405, 1895, 1991, 2668, 1721, 3025, 1891, 2882, 3045, 2755, 2255, 4136, 2737

Offset: 1

Benoit Cloitre, Apr 05 2002

Equals A127569 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 19 2007
Equals row sums of triangle A143309 and of triangle A143312. - Gary W. Adamson, Aug 06 2008
Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - R. J. Mathar, Feb 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wenchang Chu, Jordan's totient function and trigonometric sums, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).
Nikolai Osipov, Problem 12003, Amer. Math. Monthly, 124(8) (2017), p. 754.

Column 2 of A343510.
Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.
For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).

A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* Enrique Pérez Herrero, Feb 25 2012 *)
f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

for(n=1,100,print1((sumdiv(n,k,k*sigma(k)*moebius(n/k))),","))

a(n) = Sum_{d|n} d^2*phi(n/d). - Vladeta Jovovic, Jul 31 2002
a(n) = Sum_{k=1..n} gcd(n, k)^2. - Vladeta Jovovic, Aug 27 2003
Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - R. J. Mathar, Feb 03 2011
a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - Enrique Pérez Herrero, Feb 25 2012.
G.f.: Sum_{n >= 1} phi(n)*(x^n + x^(2*n))/(1 - x^n)^3 = x + 5*x^2 + 11*x^3 + 22*x^4  + .... - Peter Bala, Dec 30 2013
Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
From Peter Bala, Dec 26 2023: (Start)
For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).
It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - Peter Bala, Jan 16 2024
Sum_{k=1..n} a(k)/k ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 11 2024

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

Original entry on oeis.org

1, 13, 73, 205, 601, 949, 2353, 3277, 5905, 7813, 14521, 14965, 28393, 30589, 43873, 52429, 83233, 76765, 129961, 123205, 171769, 188773, 279313, 239221, 375601, 369109, 478297, 482365, 706441, 570349, 922561, 838861, 1060033, 1082029

Offset: 1

Benoit Cloitre, Jun 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001157, A001159, A002117, A007434, A013663, A065827.
Cf. A068963, A372962, A372963, A373007.
Cf. A057660, A084220, A372966, A373105.

with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n),n=1..40); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* G. C. Greubel, Oct 08 2018 *)
f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)

a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - Amiram Eldar, Oct 30 2022
From Peter Bala, Jan 18 2024: (Start)
a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.
Dirichlet g.f.: zeta(s) * zeta(s-4) / zeta(s-2) [Corrected by Michael Shamos, May 18 2025]. (End)
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_4(d). - Seiichi Manyama, May 18 2024

A056789 a(n) = Sum_{k=1..n} lcm(k,n)/gcd(k,n).

Original entry on oeis.org

1, 3, 10, 19, 51, 48, 148, 147, 253, 253, 606, 352, 1015, 738, 960, 1171, 2313, 1263, 3250, 1869, 2803, 3028, 5820, 2784, 6301, 5073, 6814, 5458, 11775, 4798, 14416, 9363, 11505, 11563, 14898, 9343, 24643, 16248, 19276, 14797, 33621, 14013, 38830

Offset: 1

Leroy Quet, Aug 20 2000

For prime p, a(p) = 1 + p^2*(p-1)/2.
We note lcm(k,n) = k*n iff gcd(k,n) = 1 (and in general lcm(k,n) equals k*n/gcd(k,n)), and so for these values LCM/GCD = k*n. From A023896, we have that Sum_{k=1..n-1: gcd(k,n)=1} k = n*phi(n)/2, and so Sum_{k=1..n-1: gcd(k,n)=1} k*n = n * Sum_{k=1..n-1: gcd(k,n)=1} k = n^2*phi(n)/2. As this is true, certainly Sum_{k=1..n} lcm(k,n)/gcd(k,n) > n^2*phi(n)/2. - Jon Perry, Nov 09 2014 [Edited by Petros Hadjicostas, May 27 2020]
Conjecture: for prime p, a(p^n) = 1 + (1/2)*(p - 1)*p^2*(p^(3*n) - 1)/(p^3 - 1) for n = 1,2,3,.... Cf. A339384. - Peter Bala, Dec 04 2020
The conjecture can be proven by splitting up the sum like this: a(p^n) = 1 + Sum_{1 <= r < p^n if gcd(p,r) = 1} lcm(p^n,r)/gcd(p^n,r) + Sum_{1 <= r < p^(n-1) if gcd(p,r) = 1} lcm(p^n,p*r)/gcd(p^n,p*r) + … + Sum_{1 <= r < p if gcd(p,r) = 1} lcm(p^n,p^(n-1)*r)/gcd(p^n,p^(n-1)*r) = 1 + Sum_{1 <= r < p^n if gcd(p,r) = 1} p^n*r + Sum_{1 <= r < p^(n-1) if gcd(p,r) = 1} p^(n-1)*r + … + Sum_{1 <= r < p if gcd(p,r) = 1} p*r = 1 + p^n*(1/2)*p^n*phi(p^n) +  p^(n-1)*(1/2)*p^(n-1)*phi(p^(n-1)) + … + p*(1/2)*p*phi(p) = 1 + (1/2)*(p-1)*Sum_{k=1..n} p^(3k-1) = 1 + (1/2)*(p-1)*p^2*(p^(3*n)-1)/(p^3-1). - Sebastian Karlsson, Dec 07 2020

			a(6) = 6/1 + 6/2 + 6/3 + 12/2 + 30/1 + 6/6 = 48.

T. D. Noe, Table of n, a(n) for n = 1..1000

Row sums of triangle in A051537.

Cf. A023896, A068963, A339384.

a056789 = sum . a051537_row  -- Reinhard Zumkeller, Jul 07 2013

Table[ Sum[ LCM[k, n] / GCD[k, n], {k, 1, n}], {n, 1, 50}]
f[p_, e_] := p^2*(p-1)*(p^(3*e)-1)/(p^3-1)+1; a[1] = 1; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 40] (* Amiram Eldar, Oct 05 2023 *)

vector(50, n, sum(k=1, n, lcm(k,n)/gcd(k,n))) \\ Michel Marcus, Nov 08 2014

a(n) = sumdiv(n, d, if(d>1, d^2*eulerphi(d)/2, 1)); \\ Daniel Suteu, Dec 10 2020

a(n) > n^2*phi(n)/2. - Thomas Ordowski, Nov 08 2014
a(n) = Sum_{k=1..n} k*n/gcd(k,n)^2. - Thomas Ordowski, Nov 08 2014
a(n) = (1/2)*Sum_{d|n} d^2*(d+1) Sum_{j|n/d} mu(j)*j^2. - Felix A. Pahl, Nov 23 2019
a(n) = 1 + Sum_{d|n, d > 1} phi(d^3)/2. - Daniel Suteu, Dec 10 2020
From Amiram Eldar, Oct 05 2023: (Start)
a(n) = (A068963(n)+1)/2.
Sum_{k=1..n} a(k) ~ (Pi^2/120) * n^4. (End)
a(n) < n^3 / 2, n > 1. - Bill McEachen, Jul 18 2024
Hence n^3/log log n << a(n) << n^3. - Charles R Greathouse IV, Jul 25 2024

Additional comments from Amarnath Murthy, May 09 2002

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 29, 235, 925, 3101, 6815, 16759, 29597, 57097, 89929, 160931, 217375, 371125, 486011, 728735, 947101, 1419569, 1655813, 2475739, 2868425, 3938365, 4666999, 6435815, 6955295, 9690601, 10762625, 13874563, 15502075, 20510309, 21133315, 28628191, 30307229, 37818785

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A068963, A084218, A372963.
Cf. A001160, A008683.
Cf. A013662, A013664, A088246.
Cf. A350156, A372964.
Cf. A371492, A372952.

f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1); 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)*(n/d)^2*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

A068970 a(n) = Sum_{d|n} phi(d^4).

Original entry on oeis.org

1, 9, 55, 137, 501, 495, 2059, 2185, 4429, 4509, 13311, 7535, 26365, 18531, 27555, 34953, 78609, 39861, 123463, 68637, 113245, 119799, 267675, 120175, 313001, 237285, 358723, 282083, 682893, 247995, 893731, 559241, 732105, 707481, 1031559, 606773, 1823509

Offset: 1

Benoit Cloitre, Apr 06 2002

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

Cf. A000010, A013661, A013663, A057660, A068963, A189393.

f:= n -> add(numtheory:-phi(d^4),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Sep 13 2018

Table[Total[EulerPhi[Divisors[n]^4]], {n, 40}] (* Vincenzo Librandi, Sep 13 2018 *)
f[p_, e_] := 1 + p^3*(p - 1)*(p^(4*e) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Dec 01 2022 *)

a(n) = sumdiv(n, d, eulerphi(d^4)); \\ Michel Marcus, Mar 10 2018

Also Sum_{d|n} d^m*phi(d^(4-m)) for m=0, 1, 2, 3.
Multiplicative with a(p^e) = 1 + p^3 * (p-1)(p^(4e)-1)/(p^4-1).
G.f.: Sum_{k>=1} k^3*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Dec 01 2022
From Peter Bala, Jan 21 2024: (Start)
a(n) = Sum_{k = 1..n} (n/gcd(k, n))^3 = Sum_{k = 1..n} (lcm(k, n)/k)^3.
Dirichlet g.f.: zeta(s) * zeta(s-4)/zeta(s-3). (End)

A321349 a(n) = Sum_{d|n} phi(d^n), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 19, 137, 2501, 16071, 705895, 8421505, 258293449, 4007813013, 259374246011, 2972767821815, 279577021469773, 4762869973595499, 233543432626753439, 9223512776490647553, 778579070010669895697, 13115569455375954492093, 1874292305362402347591139

Offset: 1

Ilya Gutkovskiy, Nov 06 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000010, A057660, A068963, A068970, A226459, A226561.

Table[Sum[EulerPhi[d^n], {d, Divisors[n]}], {n, 19}]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k - 1) EulerPhi[k] x^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(n/GCD[n, k])^(n - 1), {k, n}], {n, 19}]

a(n) = sumdiv(n, d, eulerphi(d^n)); \\ Michel Marcus, Nov 06 2018

G.f.: Sum_{k>=1} k^(k-1)*phi(k)*x^k/(1 - (k*x)^k).
a(n) = Sum_{d|n} d^(n-1)*phi(d).
a(n) = Sum_{k=1..n} (n/gcd(n,k))^(n-1).
From Richard L. Ollerton, May 08 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)^n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^(n-1)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.

Original entry on oeis.org

1, 61, 721, 3901, 15601, 43981, 117601, 249661, 525601, 951661, 1771441, 2812621, 4826641, 7173661, 11248321, 15978301, 24137281, 32061661, 47045521, 60859501, 84790321, 108057901, 148035361, 180005581, 243765601, 294425101, 383163121, 458761501, 594822481, 686147581

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A068963, A084218, A372962.
Cf. A084220.
Cf. A013663, A013665.

f[p_, e_] := (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1); 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)*(n/d)^2*sigma(d, 6));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_6(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(5) = 0.972439277... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A373007 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^2.

Original entry on oeis.org

1, 125, 2179, 15997, 78101, 272375, 823495, 2047613, 4765465, 9762625, 19487051, 34857463, 62748349, 102936875, 170182079, 262094461, 410338385, 595683125, 893871379, 1249381697, 1794395605, 2435881375, 3404824919, 4461748727, 6101640601, 7843543625, 10422071947

Offset: 1

Seiichi Manyama, May 25 2024

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

Cf. A371491, A371878, A373103, A373105.
Cf. A068963, A084218, A372962, A372963.
Cf. A013664, A013666.

f[p_, e_] := (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 27] (* Amiram Eldar, May 25 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 7));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_7(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(6) = Pi^2/10 = 0.986960440108... . (End)

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.

cp's OEIS Frontend

A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A069097 Moebius transform of A064987, n*sigma(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A056789 a(n) = Sum_{k=1..n} lcm(k,n)/gcd(k,n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A068970 a(n) = Sum_{d|n} phi(d^4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A321349 a(n) = Sum_{d|n} phi(d^n), where phi() is the Euler totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica