A051193 -id:A051193 - cp's OEIS frontend

A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

1, 3, 7, 11, 21, 21, 43, 43, 61, 63, 111, 77, 157, 129, 147, 171, 273, 183, 343, 231, 301, 333, 507, 301, 521, 471, 547, 473, 813, 441, 931, 683, 777, 819, 903, 671, 1333, 1029, 1099, 903, 1641, 903, 1807, 1221, 1281, 1521, 2163, 1197, 2101, 1563, 1911, 1727

Offset: 1

Henry Gould, Oct 15 2000

Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001
Also inverse Moebius transform of EulerPhi(n^2), A002618.
Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63.
a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017
From Jake Duzyk, Jun 06 2023: (Start)
These are the "contraharmonic means" of the improper divisors of square integers (inclusive of 1 and the square integer itself).
Permitting "Contraharmonic Divisor Numbers" to be defined analogously to Øystein Ore's Harmonic Divisor Numbers, the only numbers for which there exists an integer contraharmonic mean of the divisors are the square numbers, and a(n) is the n-th integer contraharmonic mean, expressible also as the sum of squares of divisors of n^2 divided by the sum of divisors of n^2. That is, a(n) = sigma_2(n^2)/sigma(n^2).
(a(n) = A001157(k)/A000203(k) where k is the n-th number such that A001157(k)/A000203(k) is an integer, i.e., k = n^2.)
This sequence is an analog of A001600 (Harmonic means of divisors of harmonic numbers) and A102187 (Arithmetic means of divisors of arithmetic numbers). (End)

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152.
H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), Vol. 39, No. 1 (1997), pp. 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), Vol. 39, No. 2 (1997), pp. 183-194.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154-5.
Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753.
Yadollah Marefat, Ali Iranmanesh, and Abolfazl Tehranian, On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.
Mathematical Reflections, Solution to Problem U338, Issue 4, 2015, p 17.
Joachim von zur Gathen, Arnold Knopfmacher, Florian Luca, Lutz G. Lucht, and Igor E. Shparlinski, Average order of cyclic groups, J. Théorie Nombres Bordeaux 16 (1) (2004) 107-123.

Cf. A000010, A000203, A000290, A001157, A018804, A050873, A051193, A054522, A057661, A061255, A065764, A078747, A174405 (partial sums), A176797, A226512.

Cf. A308471.

a057660 n = sum $ map (div n) $ a050873_row n
-- Reinhard Zumkeller, Nov 25 2013

Table[ DivisorSigma[ 2, n^2 ] / DivisorSigma[ 1, n^2 ], {n, 1, 128} ]
Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)

a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)))

a(n)=sumdivmult(n,d, eulerphi(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

from math import gcd
def A057660(n): return sum(n//gcd(n,k) for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057660(n): return prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024

a(n) = Sum_{d|n} d*A000010(d) = Sum_{d|n} d*A054522(n,d), sum of d times phi(d) for all divisors d of n, where phi is Euler's phi function.
a(n) = sigma_2(n^2)/sigma_1(n^2) = A001157(A000290(n))/A000203(A000290(n)) = A001157(A000290(n))/A065764(n). - Labos Elemer, Nov 21 2001
a(n) = Sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010
a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013
G.f.: Sum_{n >= 1} n*phi(n)*x^n/(1 - x^n) = x + 3*x^2 + 7*x^3 + 11*x^4 + .... Dirichlet g.f.: sum {n >= 1} a(n)/n^s = zeta(s)*zeta(s-2)/zeta(s-1) for Re s > 3.  Cf. A078747 and A176797. - Peter Bala, Dec 30 2013
a(n) = Sum_{i=1..n} numerator(n/i). - Wesley Ivan Hurt, Feb 26 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} lcm(n,k)/k.
a(n) = Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
From Vaclav Kotesovec, Jun 13 2021: (Start)
Sum_{k=1..n} a(k)/k ~ 3*zeta(3)*n^2/Pi^2.
Sum_{k=1..n} k^2/a(k) ~ A345294 * n.
Sum_{k=1..n} k*A000010(k)/a(k) ~ A345295 * n. (End)
Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/Pi^2. - Vaclav Kotesovec, Jun 10 2023

More terms from James Sellers, Oct 16 2000

A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

Original entry on oeis.org

1, 2, 4, 6, 11, 11, 22, 22, 31, 32, 56, 39, 79, 65, 74, 86, 137, 92, 172, 116, 151, 167, 254, 151, 261, 236, 274, 237, 407, 221, 466, 342, 389, 410, 452, 336, 667, 515, 550, 452, 821, 452, 904, 611, 641, 761, 1082, 599, 1051, 782, 956, 864, 1379, 821, 1166

Offset: 1

Henry Gould, Oct 15 2000

Sum of numerators of n-th order Farey series (cf. A006842). - Benoit Cloitre, Oct 28 2002
Equals row sums of triangle A143613. - Gary W. Adamson, Aug 27 2008
Equals row sums of triangle A159936. - Gary W. Adamson, Apr 26 2009
Also row sums of triangle A164306. - Reinhard Zumkeller, Aug 12 2009

H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), 39 (1997), 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), 39 (1997), 183-194.

T. D. Noe, Table of n, a(n) for n = 1..1000
Zachary Franco, Problem 12114, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 469; A Dirichlet Series with Reduced Numerators, Solution to Problem 12114 by Tamas Wiandt, ibid., Vol. 128, No. 1 (2021), pp. 91-92.
Index entries for sequences related to lcm's.

Cf. A000010, A000217, A006842, A018804, A023896, A051193, A057660, A143613, A159936, A164306.

See A341316 for another version.

a057661 n = a051193 n `div` n  -- Reinhard Zumkeller, Jun 10 2015

[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]; // Jaroslav Krizek, Dec 28 2016

Table[Total[Numerator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *)

a(n)=sum(k=1,n,lcm(n,k))/n \\ Charles R Greathouse IV, Feb 07 2017

from math import lcm
def A057661(n): return sum(lcm(n,k)//n for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057661(n): return 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) = (1+A057660(n))/2.
a(n) = A051193(n)/n.
a(n) = Sum_{d|n} psi(d), where psi(m) = is the sum of totatives of m (A023896). - Jaroslav Krizek, Dec 28 2016
a(n) = Sum_{i=1..n} denominator(n/i). - Wesley Ivan Hurt, Feb 26 2017
G.f.: x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
If p is prime, then a(p) = T(p-1) + 1 = p(p-1)/2 + 1, where T(n) = n(n+1)/2 is the n-th triangular number (A000217). - David Terr, Feb 10 2019
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, May 29 2021
Dirichlet g.f.: zeta(s)*(1 + zeta(s-2)/zeta(s-1))/2 (Franco, 2019). - Amiram Eldar, Mar 26 2022

More terms from James Sellers, Oct 16 2000

A051173 Triangle read by rows: T(n, k) = lcm(n, k).

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 4, 12, 4, 5, 10, 15, 20, 5, 6, 6, 6, 12, 30, 6, 7, 14, 21, 28, 35, 42, 7, 8, 8, 24, 8, 40, 24, 56, 8, 9, 18, 9, 36, 45, 18, 63, 72, 9, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 132, 12

Offset: 1

Clark Kimberling

			Triangle begins (for the full array see A109042):
  [1]  1;
  [2]  2,  2;
  [3]  3,  6,  3;
  [4]  4,  4, 12,  4;
  [5]  5, 10, 15, 20,  5;
  [6]  6,  6,  6, 12, 30,  6;
  [7]  7, 14, 21, 28, 35, 42,  7;
  [8]  8,  8, 24,  8, 40, 24, 56,  8;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Keith Bourque and Steve Ligh, Matrices associated with multiplicative functions, Linear Algebra and its Applications 216 (1995) 267-275.
Shaofang Hong and Raphael Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Mathematical Journal, Vol. 46, No. 3 (2004) 551-569.
Eric Weisstein's World of Mathematics, Least Common Multiple
Wikipedia, Least Common Multiple
Index entries for sequences related to lcm's

Cf. A109043 (column 2), A051193 (row sums), A000384 (central terms).

Cf. A109042 (variant), A003990, A002260, A075362, A051537, A050873.

a051173 = lcm
a051173_row n = a051173_tabl !! (n-1)
a051173_tabl = map (\x -> map (lcm x) [1..x]) [1..]
-- Reinhard Zumkeller, Aug 13 2013, Jul 07 2013

A051173 := proc(u,v) ilcm(u,v) ; end proc: # R. J. Mathar, Apr 07 2011

Table[LCM[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

T(n,k) = lcm(n,k);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print;) \\ Michel Marcus, Jul 10 2017

T(n, 1) = T(n, n) = n. T(n, 2) = A109043(n). - R. J. Mathar, Apr 07 2011
T(n, k) = A075362(n, k)/A050873(n, k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n, k) = A051537(n, k) * A050873(n, k). - Reinhard Zumkeller, Jul 07 2013

A064951 a(n) = Sum_{1 <= x, y <= n} lcm(x, y).

Original entry on oeis.org

1, 7, 28, 72, 177, 303, 604, 948, 1497, 2127, 3348, 4272, 6313, 8119, 10324, 13060, 17701, 20995, 27512, 32132, 38453, 45779, 57440, 64664, 77689, 89935, 104704, 117948, 141525, 154755, 183616, 205472, 231113, 258959, 290564, 314720, 364041

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

a(n) is also the entrywise 1-norm of the n X n LCM matrix.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A018806, A051193, A057660.

Table[nn = n;Total[Level[Table[Table[LCM[i, j], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 37}] (* Geoffrey Critzer, Jan 14 2015 *)

{ a=0; for (n=1, 1000, a+=n*sum(k=1, n, n/gcd(n, k)); write("b064951.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 01 2009

a(n) = a(n-1) + 2*A051193(n) - n = a(n-1) + n*A057660(n) = Sum_{1 <= i <= j <= n} (j^2/gcd(i, j)). - Henry Bottomley, Oct 29 2001

a(n) ~ 3 * zeta(3) * n^4 / (2*Pi^2). - Vaclav Kotesovec, May 29 2021

A344508 a(n) = Sum_{k=1..n} k * lcm(k,n).

Original entry on oeis.org

1, 6, 24, 64, 175, 270, 686, 928, 1647, 2150, 4356, 3792, 8619, 8526, 11250, 14592, 25721, 19926, 40432, 31200, 44835, 53966, 87814, 58272, 108125, 106470, 132678, 124656, 224547, 132750, 294066, 232960, 284229, 316166, 372400, 291168, 600991, 496014, 560742, 484000, 909421, 531846

Offset: 1

Seiichi Manyama, May 21 2021

nonn

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

Cf. A051193, A057660, A344509, A344510.

a[n_] := Sum[k * LCM[k, n], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 21 2021 *)

PARI
```
a(n) = sum(k=1, n, k*lcm(k, n));
```

Sum_{k=1..n} a(k) ~ 2 * zeta(3) * n^5 / (5*Pi^2). - Vaclav Kotesovec, May 29 2021

A072109 Numbers k such that Sum_{i=1..k} gcd(k,i) divides Sum_{i=1..k} lcm(k,i).

Original entry on oeis.org

1, 4, 36, 125, 469, 536, 882, 1156, 8532, 8775, 25012, 32000, 34749, 36324, 37179, 61952, 147456, 405224, 451584, 644304, 954084, 1185921, 1560546, 1562500, 1982464, 3080025, 5229378, 5784025, 6138868, 9231327, 12806144, 22108500, 25509168, 25562264, 29762208, 40894464, 45001899, 47397636, 49242375

Offset: 1

Benoit Cloitre, Jun 19 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A018804, A051193.

with(numtheory): for n from 1 to 10^6 do a := divisors(n): s1 := add(a[m]*phi(a[m]),m=1..nops(a)): s2 := add(phi(a[m])/a[m],m=1..nops(a)): if type((s1+1)/(2*s2),integer) then printf(`%d,`,n); fi: od:

f[n_] := (k = n; While[ !IntegerQ[ Sum[ LCM[k, i], {i, 1, k}] / Sum[ GCD[k, i], {i, 1, k}]], k++ ]; k); j = 1; Do[ m = f[j]; Print[m]; j = m + 1, {n, 1, 9}]
f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := e*(p - 1)/p + 1; q[n_] := IntegerQ[(1 + Times @@ f1 @@@ (fct = FactorInteger[n]))/(2 * Times @@ f2 @@@ fct)]; Select[Range[10^5], q] (* Amiram Eldar, May 02 2023 *)

for(n=1,1156,if(sum(i=1,n,lcm(n,i))%sum(i=1,n,gcd(n,i))==0,print1(n,",")))

is(n) = {my(f = factor(n)); (1 + prod(i = 1, #f~, (f[i,1]^(2*f[i,2] + 1) + 1)/(f[i,1] + 1))) % (2*prod(i = 1, #f~, (f[i,2]*(f[i,1] - 1)/f[i,1] + 1))) == 0;} \\ Amiram Eldar, May 02 2023

Numbers k such that A018804(k) divides A051193(k).

Edited by Robert G. Wilson v, Jun 22 2002
More terms from Vladeta Jovovic, Jun 22 2002
More terms from Sean A. Irvine, Feb 01 2011
Corrected definition - Richard L. Ollerton, May 06 2021

A199806 Alternating LCM-sum: a(n) = Sum_{k=1..n} (-1)^(k-1)*lcm(k,n).

Original entry on oeis.org

1, 0, 0, 8, -5, 18, -14, 80, -9, 100, -44, 204, -65, 294, 30, 672, -119, 540, -152, 1040, 63, 1210, -230, 1752, -75, 2028, -54, 2996, -377, 2190, -434, 5440, 165, 4624, 280, 5472, -629, 6498, 234, 8800, -779, 6300, -860, 12188, 225, 11638, -1034, 14256, -245, 13000

Offset: 1

Laszlo Toth, Nov 10 2011

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7

Cf. A051193, A199084.

[&+[(-1)^(k-1)*Lcm(k,n):k in [1..n]]: n in [1..50]]; // Marius A. Burtea, Oct 02 2019

Table[Sum[(-1)^(k-1) LCM[k,n],{k,n}],{n,50}] (* Harvey P. Dale, Jan 30 2024 *)

a(n)=-sum(k=1,n,(-1)^k*lcm(k,n)) \\ Charles R Greathouse IV, Nov 10 2011

def A199806(n) : return add((-1)^(k-1)*lcm(k,n) for k in (1..n))
[A199806(n) for n in (1..50)]  # Peter Luschny, Nov 10 2011

A308457 Expansion of e.g.f. (1/(1 - x)) * Product_{k>=2} 1/(1 - x^k)^(phi(k)/2), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 3, 15, 93, 765, 6615, 73395, 855225, 11348505, 163593675, 2633729175, 44537325525, 829112008725, 16299062754975, 340762189642875, 7597436750528625, 178862527106888625, 4426363064514265875, 115222810432347993375, 3139125774622690978125

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..435

Cf. A000010, A023022, A023896, A051193, A057661, A061255.

nmax = 20; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^(EulerPhi[k]/2), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[LCM[k, j], {j, 1, k}] x^k/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Total[Numerator[Range[k]/k]] k! Binomial[n - 1, k - 1] a[n - k]/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} A057661(k)*x^k/k).
E.g.f.: exp(Sum_{k>=1} A051193(k)*x^k/k^2).
E.g.f.: d/dx ( exp(arctanh(x)) ) * Product_{k>=3} 1/(1 - x^k)^A023022(k).
a(n) ~ A * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi)^(2/3) - n - 1/12) * n^(n + 1/36) / (2^(1/9) * 3^(19/36) * (Pi*Zeta(3))^(1/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 28 2019
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A023896(k)/k). - Ilya Gutkovskiy, May 28 2019

A338798 a(n) = Sum_{k=1..n-1} lcm(lcm(n, k), lcm(n, n-k)).

Original entry on oeis.org

0, 2, 12, 28, 100, 90, 392, 408, 792, 810, 2420, 1356, 4732, 3346, 4560, 6320, 13872, 7506, 21660, 12140, 18900, 21802, 46552, 22008, 53000, 43290, 61668, 49980, 117740, 48450, 153760, 100192, 123552, 129506, 169260, 111420, 312132, 203642, 245544, 195640

Offset: 1

Sebastian Karlsson, Jan 18 2021

nonn

Sebastian Karlsson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A006580, A023900, A051193, A057661, A138421.

a[n_] := Sum[LCM[LCM[n, k], LCM[n, n - k]], {k, 1, n - 1}];
Table[a[n], {n, 1, 40}] (* Robert P. P. McKone, Jan 18 2021 *)

a(n) = sum(k=1, n-1, lcm(lcm(n, k), lcm(n, n-k))); \\ Michel Marcus, Jan 18 2021

from math import gcd
for n in range(1, 41):
    print(n*sum([k*(n-k)//(gcd(n,k)**2) for k in range(1, n)]), end=', ')

a(n) = n*Sum_{k=1..n-1} k*(n-k)/gcd(n,k)^2.
a(n) = (1/6)*n*Sum_{d|n} d*(d*phi(d) - A023900(d)).
a(p^e) = (1/6)*p^(e+1)*(p^e-1)*(p^(e+1) + p^(2*e+1) + p^2 + 2*p + 1)/(p^2 + p + 1).
a(prime(n)) = A138421(n). - Michel Marcus, Jan 20 2021

cp's OEIS Frontend

A071645 a(n) = A051193(A072109(n))/A018804(A072109(n)).

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A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

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A057661 a(n) = Sum_{k=1..n} lcm(n,k)/n.

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A051173 Triangle read by rows: T(n, k) = lcm(n, k).

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A064951 a(n) = Sum_{1 <= x, y <= n} lcm(x, y).

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A344508 a(n) = Sum_{k=1..n} k * lcm(k,n).

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A072109 Numbers k such that Sum_{i=1..k} gcd(k,i) divides Sum_{i=1..k} lcm(k,i).

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