A057661 -id:A057661 - 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

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

A164306 Triangle read by rows: T(n, k) = k / gcd(k, n), 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 3, 4, 1, 1, 1, 1, 2, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 3, 1, 5, 3, 7, 1, 1, 2, 1, 4, 5, 2, 7, 8, 1, 1, 1, 3, 2, 1, 3, 7, 4, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 1, 1, 1, 5, 1, 7, 2, 3, 5, 11, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1

Offset: 1

Reinhard Zumkeller, Aug 12 2009

Also the gcd of the coefficients of the partition polynomials (called 'De Moivre polynomials' by O'Sullivan, see link, Theorem 4.1). - Peter Luschny, Sep 20 2022

			From _Indranil Ghosh_, Feb 14 2017: (Start)
Triangle begins:
1,
1, 1,
1, 2, 1,
1, 1, 3, 1,
1, 2, 3, 4, 1,
1, 1, 1, 2, 5, 1,
1, 2, 3, 4, 5, 6, 1,
. . .
T(4,3) = 3 / gcd(3,4) = 3 / 1 = 3. (End)

Indranil Ghosh, Rows 1..120 of triangle, flattened.
Cormac O'Sullivan, De Moivre and Bell polynomials, arXiv:2203.02868 [math.CO], 2022.

Cf. A051537, A054531, A057661, A167192.

seq(seq(k / igcd(n, k), k = 1..n), n = 1..13); # Peter Luschny, Sep 20 2022

Flatten[Table[k/GCD[k,n],{n,20},{k,n}]] (* Harvey P. Dale, Jul 21 2013 *)

for(n=0,10, for(k=1,n, print1(k/gcd(k,n), ", "))) \\ G. C. Greubel, Sep 13 2017

Sum of n-th row = A057661(n).

T(n, k) = A051537(n, k)/A054531(n, k). - Reinhard Zumkeller, Oct 30 2009

A073089 a(n) = (1/2)*(4n - 3 - Sum_{k=1..n} A007400(k)).

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1

Offset: 1

Benoit Cloitre, Aug 18 2002

From Joerg Arndt, Oct 28 2013: (Start)
Sequence is (essentially) obtained by complementing every other term of A014577.
Turns (by 90 degrees) of a curve similar to the Heighway dragon which can be rendered as follows: [Init] Set n=0 and direction=0. [Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively. [Next] Set n=n+1 and goto (draw).
See the linked pdf files for two renderings of the curve. (End)

			From _Paul D. Hanna_, Oct 19 2012: (Start)
Let F(x) = x + 1/x + 1/x^3 + 1/x^7 + 1/x^15 + 1/x^31 +...+ 1/x^(2^n-1) +...
then F(x) = x + 1/(x + 1/(-x + 1/(-x + 1/(-x + 1/(x + 1/(x + 1/(-x + 1/(-x + 1/(x + 1/(-x + 1/(-x + 1/(x + 1/(x + 1/(x + 1/(-x + 1/(-x + 1/(x + 1/(-x + 1/(-x + 1/(-x + 1/(x +...+ 1/((-1)^a(n)*x +...)))))))))))))))))))))),
a continued fraction in which the partial quotients equal (-1)^a(n)*x.  (End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Joerg Arndt, pdf rendering of the curve described in comment, alternate rendering of the curve.
Index entries for characteristic functions

Cf. A007400, A073088 (the sum part here), A123725.

a(n)=if(n<2,0,if(n%8==1,a((n+1)/2),[1,-1,0,1,1,1,0,0,1,-1,0,1,1,0,0,0][(n%16)+1])) \\ Ralf Stephan

/* Using the Continued Fraction, Print 2^N terms of this sequence: */
{N=10;CF=contfrac(x+sum(n=1,N,1/x^(2^n-1)),2^N);for(n=1,2^N,print1((1-CF[n]/x)/2,", "))} \\ Paul D. Hanna, Oct 19 2012

a(n) = { if ( n<=1, return(0)); n-=1; my(v=2^valuation(n,2) ); return( (0==bitand(n, v<<1)) != (v%2) ); } \\ Joerg Arndt, Oct 28 2013

Recurrence: a(1) = a(4n+2) = a(8n+7) = a(16n+13) = 0, a(4n) = a(8n+3) = a(16n+5) = 1, a(8n+1) = a(4n+1).
G.f.: The following series has a simple continued fraction expansion:
x + Sum_{n>=1} 1/x^(2^n-1) = [x; x, -x, -x, -x, x, ..., (-1)^a(n)*x, ...]. - Paul D. Hanna, Oct 19 2012
a(n) = A014577(n-2) + A056594(n).  Conjecture: a(n) = (1 + (-1)^A057661(n - 1))/2 for all n > 1. - Velin Yanev, Feb 01 2021

A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 11, 18, 12, 24, 14, 24, 24, 23, 18, 33, 20, 36, 32, 36, 24, 48, 27, 42, 32, 48, 30, 72, 32, 45, 48, 54, 48, 66, 38, 60, 56, 72, 42, 96, 44, 72, 66, 72, 48, 92, 51, 81, 72, 84, 54, 96, 72, 96, 80, 90, 60, 144, 62, 96, 88, 88, 84, 144, 68, 108, 96, 144

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to gcd's.
Index entries for sequences related to lcm's.

Cf. A055155, A057660, A057661, A057670, A332618.

Cf. A013663, A013664.

a:= n-> add(d/igcd(d, n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 17 2020

Table[Sum[LCM[d, n/d]/d, {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1) + e/2, (p^(e + 2) - p)/(p^2 - 1) + (e + 1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

A332619(n) = sumdiv(n,d,lcm(d,n/d)/d); \\ Antti Karttunen, Nov 12 2021

a(n) = Sum_{d|n} d / gcd(d, n/d).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) + e/2 if e is even, and (p^(e+2)-p)/(p^2-1) + (e + 1)/2 if e is odd.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*zeta(6)/(8*zeta(5)) = 0.740543... . (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.

A287006 a(1) = 1; a(n+1) = Sum_{k=1..n} lcm(a(k),a(n))/a(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 12, 13, 45, 36, 32, 86, 120, 75, 177, 250, 315, 281, 1194, 726, 925, 2695, 2218, 5776, 6808, 6632, 8383, 28449, 34934, 53325, 69653, 153540, 107261, 371925, 241534, 749726, 870493, 1460599, 2623154, 3576448, 4841995, 9911297, 15119248, 19818816, 20257600, 7481107, 80326829

Offset: 1

Ilya Gutkovskiy, Aug 31 2017

nonn

			a(1) = 1;
a(2) = lcm(a(1),a(1))/a(1) = lcm(1,1)/1 = 1;
a(3) = lcm(a(1),a(2))/a(2) + lcm(a(2),a(2))/a(2) = lcm(1,1)/1 + lcm(1,1)/1 = 2;
a(4) = lcm(a(1),a(3))/a(3) + lcm(a(2),a(3))/a(3) + lcm(a(3),a(3))/a(3) = lcm(1,2)/2 + lcm(1,2)/2 + lcm(2,2)/2 = 3, etc.

Ivan Neretin, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's

Cf. A056147, A057661, A286946.

a[1] = 1; a[n_] := a[n] = Sum[LCM[a[k - 1], a[n - 1]]/a[n - 1], {k, 2, n}]; Table[a[n], {n, 48}]
a[1] = 1; a[n_] := a[n] = Sum[a[k - 1]/GCD[a[k - 1], a[n - 1]], {k, 2, n}]; Table[a[n], {n, 48}]

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

A332049 a(n) = (1/2) * Sum_{d|n, d > 1} d * phi(d).

Original entry on oeis.org

0, 1, 3, 5, 10, 10, 21, 21, 30, 31, 55, 38, 78, 64, 73, 85, 136, 91, 171, 115, 150, 166, 253, 150, 260, 235, 273, 236, 406, 220, 465, 341, 388, 409, 451, 335, 666, 514, 549, 451, 820, 451, 903, 610, 640, 760, 1081, 598, 1050, 781, 955, 863, 1378, 820, 1165

Offset: 1

Ilya Gutkovskiy, Feb 06 2020

Sum of numerators of the reduced fractions 1/n, ..., (n-1)/n. Note that if n is a prime p this is p*(p-1)/2 as all fractions are already reduced. For 1/n, ..., n/n, see A057661.

			For n = 5, fractions are 1/5, 2/5, 3/5, 4/5, sum of numerators is 10.
For n = 8, fractions are 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, sum of numerators is 21.

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

Cf. A000010, A002618, A006579, A023896, A057660, A057661.

toNums a = fmap (numerator . (% a))
toNumList a = toNums a [1..(a-1)]
sumList = sum . toNumList <$> [2..200]

[0] cat [(1/2)*&+[ d*EulerPhi(d):d in Set(Divisors(n)) diff {1}]:n in [2..60]]; // Marius A. Burtea, Feb 07 2020

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 2 to N do
  v:= d*numtheory:-phi(d)/2;
  R:= [seq(i,i=d..N,d)];
  V[R]:= V[R] +~ v
od:
convert(V,list); # Robert Israel, Feb 07 2020

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

a(n) = sumdiv(n, d, if (d>1, d*eulerphi(d)))/2; \\ Michel Marcus, Feb 07 2020

G.f.: (1/2) * Sum_{k>=2} phi(k^2) * x^k / (1 - x^k).
a(n) = Sum_{k=1..n-1} k / gcd(n,k).
a(n) = (sigma_2(n^2) - sigma_1(n^2)) / (2 * sigma_1(n^2)).
a(n) = Sum_{d|n, d > 1} A023896(d).
a(n) = A057661(n) - 1 = (A057660(n) - 1) / 2.

cp's OEIS Frontend

A280133 Partial products of A057661 (Sum_{d|n} psi(d)).

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

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

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A164306 Triangle read by rows: T(n, k) = k / gcd(k, n), 1 <= k <= n.

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A073089 a(n) = (1/2)*(4n - 3 - Sum_{k=1..n} A007400(k)).

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A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

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

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