A069290 -id:A069290 - cp's OEIS frontend

A037213 Expansion of Sum_{n>=0} n*q^(n^2).

0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. - Mitch Harris, Jun 09 2005
a(n) is the square root of n if n is square, zero otherwise. - Carl R. White, May 23 2009
Möbius transform of A069290(n). - Wesley Ivan Hurt, Jul 09 2025

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000196, A008683 (mu), A010052, A066839, A069290, A070039.

a037213 n = if n == r ^ 2 then r else 0  where r = a000196 n
a037213_list = zipWith (*) a010052_list a000196_list
-- Reinhard Zumkeller, Nov 09 2012

Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 0, 100}] (* Geoffrey Critzer, Feb 21 2015 *)

a(n) = if (issquare(n), sqrtint(n), 0); \\ Michel Marcus, Aug 21 2025

Dirichlet generating function: zeta(2*s-1). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = sqrt(n) * floor( cos^2(Pi * sqrt(n)) ). - Carl R. White, May 23 2009
a(n) = A000196(n) * A010052(n). - Reinhard Zumkeller, Jan 27 2010
Sum_{k=1..n} a(k) ~ n/2. - Vaclav Kotesovec, Aug 19 2021
a(n) = A066839(n) - A070039(n). - Ridouane Oudra, Jun 24 2025
a(n) = Sum_{d|n} A069290(d) * mu(n/d). - Wesley Ivan Hurt, Jul 09 2025

A055155 a(n) = Sum_{d|n} gcd(d, n/d).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 5, 4, 2, 8, 2, 4, 4, 10, 2, 10, 2, 8, 4, 4, 2, 12, 7, 4, 8, 8, 2, 8, 2, 14, 4, 4, 4, 20, 2, 4, 4, 12, 2, 8, 2, 8, 10, 4, 2, 20, 9, 14, 4, 8, 2, 16, 4, 12, 4, 4, 2, 16, 2, 4, 10, 22, 4, 8, 2, 8, 4, 8, 2, 30, 2, 4, 14, 8, 4, 8, 2, 20, 17, 4, 2, 16, 4, 4, 4, 12, 2, 20, 4, 8, 4, 4

Offset: 1

Leroy Quet, Jul 02 2000

a(n) is odd iff n is odd square. - Vladeta Jovovic, Aug 27 2002
From Robert Israel, Dec 26 2015: (Start)
a(n) >= A000005(n), with equality iff n is squarefree (i.e., is in A005117).
a(n) = 2 iff n is prime. (End)

			a(9) = gcd(1,9) + gcd(3,3) + gcd(9,1) = 5, since 1, 3, 9 are the positive divisors of 9.

Robert Israel, Table of n, a(n) for n = 1..10000
Ekkehard Krätzel, Werner Nowak and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., Vol. 10, No. 2 (2012), pp. 761-774.
Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Central European Journal of Mathematics, Vol. 11, No. 3 (2013), pp. 477-486, preprint, arXiv: 1204.1146 [math.NT], 2012.
Project Euler, Problem 530 - GCD of Divisors.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias and P. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, N.Y., 2014, pp. 483-514, arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000005, A000188, A001620, A005117, A057670, A073002. A082633, A201994.

Cf. A000010.

N:= 1000: # to get a(1) to a(N)
V:= Vector(N):
for k from 1 to N do
   for j from 1 to floor(N/k) do
     V[k*j]:= V[k*j]+igcd(k,j)
   od
od:
convert(V,list); # Robert Israel, Dec 26 2015

Table[DivisorSum[n, GCD[#, n/#] &], {n, 94}] (* Michael De Vlieger, Sep 23 2017 *)
f[p_, e_] := If[EvenQ[e], (p^(e/2)*(p+1)-2)/(p-1), 2*(p^((e+1)/2)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

a(n) = sumdiv(n, d, gcd(d, n/d)); \\ Michel Marcus, Aug 03 2016

from sympy import divisors, gcd
def A055155(n): return sum(gcd(d,n//d) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 19 2021

Multiplicative with a(p^e) = (p^(e/2)*(p+1)-2)/(p-1) for even e and a(p^e) = 2*(p^((e+1)/2)-1)/(p-1) for odd e. - Vladeta Jovovic, Nov 01 2001
Dirichlet g.f.: (zeta(s))^2*zeta(2s-1)/zeta(2s); inverse Mobius transform of A000188. - R. J. Mathar, Feb 16 2011
Dirichlet convolution of A069290 and A008966. - R. J. Mathar, Oct 31 2011
Sum_{k=1..n} a(k) ~ 3*n / (2*Pi^6) * (Pi^4 * log(n)^2 + ((8*g - 2)*Pi^4 - 24 * Pi^2 * z1) * log(n) + 2*Pi^4 * (1 - 4*g + 5*g^2 - 6*sg1) + 288 * z1^2 - 24 * Pi^2 * (-z1 + 4*g*z1 + z2)), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 01 2019
a(n) = (1/n)*Sum_{i=1..n} sigma(gcd(n,i^2)). - Ridouane Oudra, Dec 30 2020
a(n) = Sum_{k=1..n} gcd(gcd(n,k),n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010. - Richard L. Ollerton, May 09 2021

A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 28, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 56, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 84, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 120, 65, 66, 67

Offset: 1

Werner Schulte, Jan 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A069290, A055615, A037213.
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =
0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 20 2021

f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2021 *)

A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ Antti Karttunen, Aug 19 2021

Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).
Dirichlet convolution of A000010 and A069290.
Dirichlet convolution with A055615 equals A037213.
G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021
Sum_{k=1..n} a(k) ~ zeta(3)*n^2/2. - Vaclav Kotesovec, Aug 19 2021
a(n) = n * Sum_{d^2|n} 1/d. - Wesley Ivan Hurt, Feb 14 2022

A069291 Number of square divisors of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Terms 1, 2, 3, ... occurs for the first time at 1, 16, 108, 288, 1296, 3600, 10368, 14400, ... - Antti Karttunen, Nov 20 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000188, A035316, A046951, A069290, A069292, A069293.

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 120}] (* Michael De Vlieger, Nov 20 2017 *)

A069291(n) = sumdiv(n, d, (issquare(d)&&((d^2)<=n))); \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 04 2020

More terms from Antti Karttunen, Nov 20 2017

A069293 Sum of square divisors of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 5, 1

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000188, A035316, A046951, A069290, A069291, A069292.

Table[DivisorSum[n, # &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 105}] (* Michael De Vlieger, Nov 20 2017 *)

A069293(n) = sumdiv(n, d, (issquare(d)&&((d^2)<=n))*d); \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} k^2 * x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 04 2020

More terms from Antti Karttunen, Nov 20 2017

A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of cube roots of cube divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Atul Dixit, Bibekananda Maji, and Akshaa Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=3,n).

Cf. A000203, A053150, A061704, A069290, A113061, A333844.

nmax = 108; CoefficientList[Series[Sum[k x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/3) &, IntegerQ[#^(1/3)] &], {n, 108}]
f[p_, e_] := (p^(Floor[e/3] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

a(n) = {my(f = factor(n)); prod(i=1, #f~, (f[i,1]^(f[i,2]\3 + 1) - 1)/(f[i,1] - 1));} \\ Amiram Eldar, Sep 05 2023

Dirichlet g.f.: zeta(s) * zeta(3*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/3) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ Pi^2*n/6 + zeta(2/3)*n^(2/3)/2. - Vaclav Kotesovec, Dec 01 2020
a(n) = A000203(A053150(n)) (the sum of divisors of the cube root of largest cube dividing n). - Amiram Eldar, Sep 05 2023

A069292 Sum of square roots of square divisors of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A035316, A046951, A000188, A069290, A069291, A069293.

Table[DivisorSum[n, Sqrt@ # &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 105}] (* Michael De Vlieger, Nov 20 2017 *)

A069292(n) = { my(r="NA"); sumdiv(n, d, (issquare(d,&r)&&((d^2)<=n))*r); } \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 19 2021

More terms from Antti Karttunen, Nov 20 2017

A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 8, 6, 4, 2, 10, 2, 4, 4, 15, 2, 12, 2, 10, 4, 4, 2, 16, 8, 4, 10, 10, 2, 8, 2, 22, 4, 4, 4, 30, 2, 4, 4, 16, 2, 8, 2, 10, 12, 4, 2, 30, 10, 16, 4, 10, 2, 20, 4, 16, 4, 4, 2, 20, 2, 4, 12, 37, 4, 8, 2, 10, 4, 8, 2, 48, 2, 4, 16, 10, 4, 8, 2, 30, 23, 4, 2, 20, 4, 4, 4, 16, 2, 24

Offset: 1

Robert G. Wilson v, Sep 30 2006

Apparently multiplicative and the inverse Mobius transform of A069290. - R. J. Mathar, Feb 07 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ekkehard Krätzel, Werner Georg Nowak, and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Cent. Eur. J. Math., Vol. 11, No. 3 (2013), pp. 477-486; arXiv preprint, arXiv:1204.1146 [math.NT], 2012.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias, P. Pardalos (eds.) Mathematics Without Boundaries, Springer, New York, NY, 2024; arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000203, A001616, A001620, A069290, A061884, A082633, A124315.

A124316 := proc(n) local a,d;
  a := 0 ;
  for d in numtheory[divisors](n) do
     igcd(d,n/d) ;
     a := a+numtheory[sigma](%) ;
   end do:
   a;
end proc: # R. J. Mathar, Apr 14 2011

Table[Plus @@ Map[DivisorSigma[1, GCD[ #, n/# ]] &, Divisors@n], {n, 90}]
f[p_, e_] := (If[OddQ[e], 2*p^((e+3)/2), p^(e/2 + 1)*(p+1)] - (e+3)*p + e + 1)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 28 2024 *)

a(n) = sumdiv(n, d, sigma(gcd(d, n/d))); \\ Michel Marcus, Feb 13 2016

from sympy import divisors, divisor_sigma, gcd
def a(n): return sum([divisor_sigma(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

From Amiram Eldar, Mar 28 2024: (Start)
Multiplicative with a(p^e) = (p^(e/2 + 1)*(p+1) - (e+3)*p + e + 1)/(p-1)^2, if e is even, and (2*p^((e+3)/2) - (e+3)*p + e + 1)/(p-1)^2 if e is odd.
Dirichlet g.f.: zeta(s)^2 * zeta(2*s-1).
Sum_{k=1..n} a(k) = (log(n)^2/4 + (2*gamma - 1/2)*log(n) + 5*gamma^2/2 - 2*gamma - 3*gamma_1 + 1/2) * n + O(n^(2/3)*log(n)^(16/9)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633) (Krätzel et al., 2012). (End)

A333844 G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of 4th roots of 4th powers dividing n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=1,k=4,n).

Cf. A000203, A053164, A063775, A069290, A300909, A333843.

nmax = 112; CoefficientList[Series[Sum[k x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/4) &, IntegerQ[#^(1/4)] &], {n, 112}]
f[p_, e_] := (p^(Floor[e/4] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

Dirichlet g.f.: zeta(s) * zeta(4*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/4) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/2)*sqrt(n)/2. - Vaclav Kotesovec, Dec 01 2020

A344128 a(n) = Sum_{k=1..n} k * floor(n/k^2).

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 12, 16, 17, 18, 21, 22, 23, 24, 31, 32, 36, 37, 40, 41, 42, 43, 46, 52, 53, 57, 60, 61, 62, 63, 70, 71, 72, 73, 85, 86, 87, 88, 91, 92, 93, 94, 97, 101, 102, 103, 110, 118, 124, 125, 128, 129, 133, 134, 137, 138, 139, 140, 143, 144, 145, 149, 164, 165, 166

Offset: 1

Wesley Ivan Hurt, May 09 2021

nonn

Partial sums of A069290.

Cf. A013936.

b:= n-> mul((i[1]^(iquo(i[2], 2)+1)-1)/(i[1]-1), i=ifactors(n)[2]):
a:= proc(n) a(n):= a(n-1)+b(n) end: a(0):=0:
seq(a(n), n=1..66);  # Alois P. Heinz, May 14 2021

Table[Sum[k*Floor[n/k^2], {k, n}], {n, 100}]

G.f.: (1/(1 - x)) * Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, May 14 2021

cp's OEIS Frontend

A037213 Expansion of Sum_{n>=0} n*q^(n^2).

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A055155 a(n) = Sum_{d|n} gcd(d, n/d).

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A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

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A069291 Number of square divisors of n <= sqrt(n).

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A069293 Sum of square divisors of n <= sqrt(n).

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A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).

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A069292 Sum of square roots of square divisors of n <= sqrt(n).

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