A113061 -id:A113061 - cp's OEIS frontend

A035316 Sum of the square divisors of n.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1

Offset: 1

N. J. A. Sloane

The Dirichlet generating function is zeta(s)*zeta(2s-2). The sequence is the Dirichlet convolution of A000012 with the sequence defined by n*A010052(n). - R. J. Mathar, Feb 18 2011

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Nick Hobson, Table of n, a(n) for n = 1..1000
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=2,k=2,n).
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A001157, A010052, A027748, A124010, A113061 (sum cube divs).
Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), this sequence (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A008836, A000203.

a035316 n = product $
   zipWith (\p e -> (p ^ (e + 2 - mod e 2) - 1) `div` (p ^ 2 - 1))
           (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 28 2014

A035316 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        if type(e,'even') then
            e := e+2 ;
        else
            e := e+1 ;
        end if;
        a := a*(p^e-1)/(p^2-1) ;
    end do:
    a ;
end proc:
seq(A035316(n),n=1..100) ; # R. J. Mathar, Oct 10 2017

Table[ Plus @@ Select[ Divisors@ n, IntegerQ@ Sqrt@ # &], {n, 93}] (* Robert G. Wilson v, Feb 19 2011 *)
f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

vector(93, n, sumdiv(n, d, issquare(d)*d))

a(n)=my(f=factor(n));prod(i=1,#f[,1],(f[i,1]^(f[i,2]+2-f[i,2]%2)-1)/(f[i,1]^2-1)) \\ Charles R Greathouse IV, May 20 2013

Multiplicative with a(p^e)=(p^(e+2)-1)/(p^2-1) for even e and a(p^e)=(p^(e+1)-1)/(p^2-1) for odd e. - Vladeta Jovovic, Dec 05 2001
G.f.: Sum_{k>0} k^2*x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n^2) = A001157(n). - Michel Marcus, Jan 14 2014
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Sum_{k=1..n} a(k) ~ Zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Feb 04 2019
a(n) = Sum_{k=1..n} k * (floor(sqrt(k)) - floor(sqrt(k-1))) * (1 - ceiling(n/k) + floor(n/k)). - Wesley Ivan Hurt, Jun 13 2021
a(n) = Sum_{d|n} d * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d*sigma(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A380396 a(n) is the sum of the unitary divisors of n that are cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 23 2025

The number of unitary divisors of n that are cubes is A380395(n).

			a(8) = 9 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3, and 1 + 8 = 9.
a(216) = 252 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3, and 1 + 8 + 27 + 216 = 252.

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

Cf. A000578, A046100, A077610, A113061, A358347, A366761, A371334, A380395.

f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, 1, f[i, 1]^f[i, 2] + 1));}

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * [d is cube], where [] is the Iverson bracket.
a(n) >= 1, with equality if and only if n is not in A366761.
a(n) <= A113061(n), with equality if and only if n is biquadratefree (A046100).
Multiplicative with a(p^e) = p^e + 1 if e is divisible by 3, and 1 otherwise.
Sum_{k=1..n} a(k) ~ c * n^(4/3) / 4, where c = zeta(4/3)/zeta(7/3) = 2.54455250463133711749... .
Dirichlet g.f.: zeta(s) * zeta(3*s-3) / zeta(4*s-3).
In general, the average order of the sum of the unitary divisors that are m-powers is c * n^(1+1/m) / (m+1), where c = zeta(1+1/m)/zeta(2+1/m), and its Dirichlet g.f. is zeta(s) * zeta(m*s-m) / zeta((m+1)*s-m), both for m >= 2.

A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 36, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 73, 65, 66, 67, 68, 69, 70, 71, 81, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is a cube.

Inverse Moebius transform of A078429.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000578, A004709 (fixed points), A010057, A061704, A076752, A078429, A113061.

nmax = 75; CoefficientList[Series[Sum[x^(k^3)/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[(n/#)^(1/3)] &]; Table[a[n], {n, 1, 75}]
f[p_, e_] := (p^(e+3) - p^Mod[e, 3])/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

A327626(n) = sumdiv(n,d,ispower(n/d,3)*d); \\ Antti Karttunen, Sep 19 2019

a(n) = Sum_{d|n} A078429(d).
a(n) = Sum_{d|n} A010057(n/d) * d. Dirichlet convolution of A000027 and A010057.
D.g.f.: zeta(s-1)*zeta(3s). - R. J. Mathar, Jun 05 2020
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1890. - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = (p^(e+3) - p^(e mod 3))/(p^3-1). - Amiram Eldar, May 25 2025

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

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A300909 Sum of 4th powers dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset: 1

Ilya Gutkovskiy, Mar 15 2018

Multiplicative with a(p^e) = (p^(4*(1+floor(e/4)))-1)/(p^4-1). - Robert Israel, Mar 15 2018

			a(16) = 17 because 16 has 5 divisors {1, 2, 4, 8, 16} among which 2 divisors {1, 16} are 4th powers and 1 + 16 = 17.
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^10/10 + x^11/11 + x^12/12 + x^13/13 + x^14/14 + x^15/15 + 17*x^16/16 + x^17/17 + ...
exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + 2*x^16 + 2*x^17 + ... + A046042(n)*x^n + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
A. Dixit, B. Maji, A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=4,k=4,n).

Cf. A000583, A001159, A035316, A046042, A046100 (positions of ones), A063775, A113061.

N:= 1000: # for a(1)..a(N)
V:= Vector(N,1):
for m from 2 to floor(N^(1/4)) do
  R:= [seq(i,i=m^4 .. N, m^4)];
  V[R]:= map(`+`,V[R],m^4)
od:
convert(V,list); # Robert Israel, Mar 15 2018

Table[DivisorSum[n, # &, IntegerQ[#^(1/4)] &], {n, 112}]
nmax = 112; Rest[CoefficientList[Series[Sum[k^4 x^k^4/(1 - x^k^4), {k, 1, 10}], {x, 0, nmax}], x]]
nmax = 112; Rest[CoefficientList[Series[-Log[Product[(1 - x^k^4), {k, 1, 10}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p^(4*(1 + Floor[e/4])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

a(n) = sumdiv(n, d, d*ispower(d, 4)); \\ Michel Marcus, Mar 15 2018

G.f.: Sum_{k>=1} k^4*x^(k^4)/(1 - x^(k^4)).
L.g.f.: -log(Product_{k>=1} (1 - x^(k^4))) = Sum_{n>=1} a(n)*x^n/n.
D.g.f.: zeta(s)*zeta(4s-4). - Robert Israel, Mar 15 2018
Sum_{k=1..n} a(k) ~ zeta(5/4)*n^(5/4)/5 - n/2. - Vaclav Kotesovec, Dec 01 2020

A309126 a(n) = n + 2^3 * floor(n/2^3) + 3^3 * floor(n/3^3) + 4^3 * floor(n/4^3) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 78, 79, 80, 81, 82, 91, 92, 93, 94, 95, 96, 97, 98, 107, 108, 109, 110, 111, 112, 113, 114, 123, 124, 125, 126, 127, 128, 156, 157, 166, 167, 168, 169, 170, 171, 172, 173, 246, 247, 248, 249, 250, 251, 252

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

nonn

Partial sums of A113061.

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

Cf. A013937, A024916, A113061, A309125, A309127.

Table[Sum[k^3 Floor[n/k^3], {k, 1, n}], {n, 1, 70}]
nmax = 70; CoefficientList[Series[1/(1 - x) Sum[k^3 x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, k^3*(n\k^3)); \\ Seiichi Manyama, Aug 30 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} k^3 * x^(k^3)/(1 - x^(k^3)).

a(n) ~ zeta(4/3)*n^(4/3)/4 - n/2. - Vaclav Kotesovec, Aug 30 2021

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A304906 L.g.f.: log(Product_{k>=1} (1 + x^(k^3))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 28, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -28, 1, 7, 1, -1, 1, -1, 1, -1, 1, 55, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 28, -1

Offset: 1

Ilya Gutkovskiy, May 20 2018

			L.g.f.: L(x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 + 7*x^8/8 + x^9/9 - x^10/10 + x^11/11 - x^12/12 + ...
exp(L(x)) = 1 + x + x^8 + x^9 + x^27 + x^28 + x^35 + x^36 + x^64 + x^65 + x^72 + x^73 + ... + A279329(n)*x^n + ...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors.

Cf. A000578, A010057, A004709, A113061, A279329, A300853.

nmax = 82; Rest[CoefficientList[Series[Log[Product[1 + x^k^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 82; Rest[CoefficientList[Series[Sum[k^3 x^k^3/(1 + x^k^3), {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/3)] &], {n, 82}]
f[p_, e_] := (-1)^(p+1) * (p^(3*Floor[(e+3)/3])-1)/(p^3-1) + If[p == 2 && Divisible[e, 3], 2^(e + 1), 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)

seq(n)={Vec(deriv(log(prod(k=1, sqrtnint(n, 3), 1 + x^(k^3) + O(x*x^n)))))} \\ Andrew Howroyd, Jul 20 2018

a(n)={sumdiv(n, d, if(n%d^3, 0, (-1)^(n/d^3 + 1) * d^3))} \\ Andrew Howroyd, Jul 20 2018

G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 + x^(k^3)).
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010057(d)*d.
a(n) = 1 if n is an odd cubefree.
a(n) = -1 if n is an even cubefree.
Multiplicative with a(p^e) = (-1)^(p+1) * (p^(3*floor((e+3)/3))-1)/(p^3-1) + if(p = 2 and e == 0 (mod 3), 2^(e + 1) otherwise 0). - Amiram Eldar, Oct 03 2023

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A347398 Expansion of g.f. Sum_{k>=1} k^k * x^(k^k)/(1 - x^(k^k)).

Original entry on oeis.org

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, 28, 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, 28, 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, 28, 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, 32, 1, 1, 1, 5

Offset: 1

Seiichi Manyama, Aug 30 2021

nonn

			1^1 | 108, 2^2 | 108 and 3^3 | 108. So a(108) = 1^1 + 2^2 + 3^3 = 32.

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

Cf. A000203, A000312, A035316, A113061, A300909, A347397, A347399.

PARI
```
a(n) = sum(k=1, n, (n%k^k==0)*k^k);
```

a(n) = A347397(n) - A347397(n-1) for n > 1.

a(n) = Sum_{k=1..n, k^k | n} k^k.

cp's OEIS Frontend

A035316 Sum of the square divisors of n.

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A380396 a(n) is the sum of the unitary divisors of n that are cubes.

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A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

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

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A385006 The sum of the biquadratefree divisors of n.

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A300909 Sum of 4th powers dividing n.

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A309126 a(n) = n + 2^3 * floor(n/2^3) + 3^3 * floor(n/3^3) + 4^3 * floor(n/4^3) + ...

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