A057660 -id:A057660 - cp's OEIS frontend

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

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

A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

Original entry on oeis.org

1, 4, 9, 18, 25, 36, 49, 78, 87, 100, 121, 162, 169, 196, 225, 326, 289, 348, 361, 450, 441, 484, 529, 702, 645, 676, 807, 882, 841, 900, 961, 1334, 1089, 1156, 1225, 1566, 1369, 1444, 1521, 1950, 1681, 1764, 1849, 2178, 2175, 2116, 2209, 2934, 2443, 2580

Offset: 1

Vladeta Jovovic, Jul 21 2001

If k is squarefree (cf. A005117) then A062952(k) = k^2. - Benoit Cloitre, Apr 16 2002

Inverse Möbius transform of A062354(n). - Wesley Ivan Hurt, Jul 26 2025

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

Cf. A005117, A029939, A057660, A062952.
Cf. A002117, A013661, A183699, A330523.
Cf. A062354.

f[p_, e_] := (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{d|n} phi(d)*sigma(d).
a(n) = Sum_{k=1..n} sigma(n/gcd(n, k)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A183699 * A330523 / 3. - Amiram Eldar, Oct 30 2022

A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

Original entry on oeis.org

1, 57, 703, 3641, 15501, 40071, 117307, 233017, 512461, 883557, 1770231, 2559623, 4824613, 6686499, 10897203, 14913081, 24132657, 29210277, 47039023, 56439141, 82466821, 100903167, 148023723, 163810951, 242203001, 275002941, 373584043

Offset: 1

Benoit Cloitre, Jun 21 2003

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

Cf. A001158, A013665, A013954.

Cf. A057660, A084218.

with(numtheory): a:=n->sigma[6](n^2)/sigma[3](n^2): seq(a(n),n=1..30); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* Harvey P. Dale, May 02 2012 *)
f[p_, e_] := (p^(6*e + 3) + 1)/(p^3 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^6)/sumdiv(n^2,d,d^3)

a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = 90*zeta(7)/(7*Pi^4) = 0.133093... . - Amiram Eldar, Oct 30 2022
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^3.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_6(d). (End)

A051193 a(n) = Sum_{k=1..n} lcm(n,k).

Original entry on oeis.org

1, 4, 12, 24, 55, 66, 154, 176, 279, 320, 616, 468, 1027, 910, 1110, 1376, 2329, 1656, 3268, 2320, 3171, 3674, 5842, 3624, 6525, 6136, 7398, 6636, 11803, 6630, 14446, 10944, 12837, 13940, 15820, 12096, 24679, 19570, 21450, 18080, 33661, 18984, 38872, 26884

Offset: 1

Clark Kimberling

nonn

Akshay Bansal, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Akshay Bansal, C Program.
Soichi Ikeda and Kaneaki Matsuoka, On the Lcm-Sum Function, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
Index entries for sequences related to lcm's.

Cf. A000010, A018804, A051173 (triangle whose n-th row sum is a(n)), A057660, A057661.

a051193 = sum . a051173_row  -- Reinhard Zumkeller, Feb 11 2014

a:=n->add(ilcm( n, j ), j=1..n): seq(a(n), n=1..50); # Zerinvary Lajos, Nov 07 2006

Table[Sum[LCM[k, n], {k, 1, n}], {n, 1, 39}] (* Geoffrey Critzer, Feb 16 2015 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := n * (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

a(n) = sum(k=1, n, lcm(n,k)); \\ Michel Marcus, Feb 06 2015

from math import prod
from sympy import factorint
def A051193(n): return n*(1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items())>>1) # Chai Wah Wu, Aug 05 2024

a(n) = n*(1+Sum_{d|n} d*phi(d))/2 = n*(1+A057660(n))/2 = n*A057661(n). - Vladeta Jovovic, Jun 21 2002
G.f.: x*f'(x), where f(x) = x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k) and phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~ 3 * zeta(3) * n^4 / (4*Pi^2). - Vaclav Kotesovec, May 29 2021

A062949 Multiplicative with a(p^e) = ((e+1)p^(e+1)-(e+2)p^e+1)/(p-1).

Original entry on oeis.org

1, 3, 5, 9, 9, 15, 13, 25, 23, 27, 21, 45, 25, 39, 45, 65, 33, 69, 37, 81, 65, 63, 45, 125, 69, 75, 95, 117, 57, 135, 61, 161, 105, 99, 117, 207, 73, 111, 125, 225, 81, 195, 85, 189, 207, 135, 93, 325, 139, 207, 165, 225, 105, 285, 189, 325, 185, 171, 117, 405

Offset: 1

Vladeta Jovovic, Jul 21 2001

Inverse Mobius transform of A062355.

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

Cf. A057660, A029939.

A062949 := proc(n) add(numtheory[phi](d)*numtheory[tau](d), d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Feb 09 2011

f[p_, e_] := ((e+1)*p^(e+1)-(e+2)*p^e+1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 60] (* Amiram Eldar, Jul 31 2019 *)

a(n) = Sum_{d|n} phi(d)*tau(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n, k)).
a(n) = Sum_{d|n} d*uphi(n/d), where uphi() = A047994(). - Vladeta Jovovic, Mar 16 2004

A372966 a(n) = sigma_8(n^2)/sigma_4(n^2).

Original entry on oeis.org

1, 241, 6481, 61681, 390001, 1561921, 5762401, 15790321, 42521761, 93990241, 214344241, 399754561, 815702161, 1388738641, 2527596481, 4042322161, 6975673921, 10247744401, 16983432721, 24055651681, 37346120881, 51656962081, 78310705441, 102337070401

Offset: 1

Seiichi Manyama, May 18 2024

Apparently, a(n) == 1 (mod 240). - Hugo Pfoertner, May 20 2024

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

Cf. A057660, A084218, A084220, A108223.
Cf. A008683, A013956.
Cf. A372965.

f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 20 2024 *)

PARI
```
a(n) = sigma(n^2, 8)/sigma(n^2, 4);
```

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 8));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^4.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_8(d).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e + 4) + 1)/(p^4 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-4).
Sum_{k=1..n} a(k) ~ (zeta(9)/(9*zeta(5))) * n^9. (End)

A350156 Inverse Moebius transform of A000056.

Original entry on oeis.org

1, 7, 25, 55, 121, 175, 337, 439, 673, 847, 1321, 1375, 2185, 2359, 3025, 3511, 4897, 4711, 6841, 6655, 8425, 9247, 12145, 10975, 15121, 15295, 18169, 18535, 24361, 21175, 29761, 28087, 33025, 34279, 40777, 37015, 50617, 47887, 54625, 53119, 68881, 58975, 79465, 72655, 81433, 85015

Offset: 1

Werner Schulte, Jan 19 2022

Let f be an arbitrary arithmetic function. Define the sequence a(f; n) by a(f; n) = Sum_{i=1..n, k=1..n} f(n / gcd(gcd(i,k),n)) for n > 0. Then a(f; n) equals inverse Moebius transform of f(n) * A007434(n) for n > 0; if f is multiplicative then a(f; n) is multiplicative; this sequence uses f(n) = n (see formula section).

Column k=2 of A372968.
Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.
Cf. A372962, A372964.

f[p_, e_] := p^(3*e) - (p - 1)*(p^(3*e) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Jan 19 2022 *)

from math import prod
from sympy import factorint
def A350156(n): return prod((q:=p**(3*e))-(p-1)*(q-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).
a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.
Dirichlet convolution with A000010 equals A000578.
Dirichlet convolution of A001158 and A055615.
Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - Amiram Eldar, Oct 16 2022
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2. - 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)

A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

Original entry on oeis.org

0, 0, 1, 4, 8, 20, 21, 56, 60, 96, 105, 220, 152, 364, 301, 360, 464, 816, 549, 1140, 760, 1036, 1221, 2024, 1196, 2200, 2041, 2484, 2184, 4060, 2205, 4960, 3664, 4224, 4641, 5180, 4008, 8436, 6517, 7072, 5980, 11480, 6321, 13244, 8888, 9540, 11661, 17296

Offset: 0

N. J. A. Sloane

nonn

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from Reinhard Zumkeller)
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.

Antidiagonal sums of array A003990.
Cf. A209295.
Cf. A000010, A023900, A057660, A130054.

a006580 n = a006580_list !! (n-1)
a006580_list = map sum a003990_tabl
-- Reinhard Zumkeller, Aug 05 2012

a:= n-> add(ilcm(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019

Table[ Sum[ LCM[ k, n-k ], {k, 1, n-1} ], {n, 2, 50} ] (* Olivier Gérard, Aug 15 1997 *)
f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := 1 - (p - 1)*e; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct)*n/6; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ Michel Marcus, Aug 11 2017

For n > 0, a(n) = (n/6)*Sum_{d|n} (d*phi(d) - A023900(d)). - Sebastian Karlsson, Oct 02 2021

a(n) = (n/6) * (A057660(n) - A130054(n)), for n > 0. - Amiram Eldar, Apr 28 2023

More terms from Olivier Gérard, Aug 15 1997

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)

cp's OEIS Frontend

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

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A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

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A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

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A051193 a(n) = Sum_{k=1..n} lcm(n,k).

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A062949 Multiplicative with a(p^e) = ((e+1)*p^(e+1)-(e+2)*p^e+1)/(p-1).

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A372966 a(n) = sigma_8(n^2)/sigma_4(n^2).

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A350156 Inverse Moebius transform of A000056.

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A062949 Multiplicative with a(p^e) = ((e+1)p^(e+1)-(e+2)p^e+1)/(p-1).