A035316 -id:A035316 - cp's OEIS frontend

A360159 a(n) is the sum of divisors of n that are odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 50, 26, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 91, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

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

Cf. A016754, A035316, A078434, A344300.

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); f[2, e_] := 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, 1] == 2, 1, (f[i, 1]^(2*(f[i, 2]\2)+2)-1)/(f[i, 1]^2-1))); }

a(n) = Sum_{d|n, d odd square} d.
a(n) = (A035316(n) + A344300(n))/2.
Multiplicative with a(2^e) = 1, and for p > 2, 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.
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-4^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/6 = 0.4353958914... .

A055928 Sum of square divisors of n! = sum of squares of divisors of the square root of largest square dividing n!.

Original entry on oeis.org

1, 1, 1, 5, 5, 210, 210, 850, 7735, 806806, 806806, 3229590, 3229590, 161479500, 1455090000, 23286770000, 23286770000, 838446909300, 838446909300, 83973923013750, 83973923013750, 10244818607677500, 10244818607677500

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..505

A001157, A000142, A035316, A055772, A000188.

f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p^2-1) , (p^(e+2)-1)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 23] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A035316(n!) = A001157(A055772(n)) = A001157(A000188(n!)).

A065745 Sum of squares and twice squares dividing n.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 15, 10, 3, 1, 7, 1, 3, 1, 31, 1, 30, 1, 7, 1, 3, 1, 15, 26, 3, 10, 7, 1, 3, 1, 63, 1, 3, 1, 70, 1, 3, 1, 15, 1, 3, 1, 7, 10, 3, 1, 31, 50, 78, 1, 7, 1, 30, 1, 15, 1, 3, 1, 7, 1, 3, 10, 127, 1, 3, 1, 7, 1, 3, 1, 150, 1, 3, 26, 7, 1, 3, 1, 31, 91, 3, 1, 7, 1, 3, 1, 15, 1, 30, 1

Offset: 1

Vladeta Jovovic, Dec 04 2001

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A035316, A065704, A078434.

f[2, e_] := 2^(e+1) - 1; f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p^2-1), (p^(e+2)-1)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 13 2020 *)

a(n) = sumdiv(n, d, issquare(d)*d + (1 - d%2)*issquare(d/2)*d) \\ Michel Marcus, Jun 17 2013

Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = (p^(e+2)-1)/(p-1)/(p+1) for odd p and even e and a(p^e) = (p^(e+1)-1)/(p-1)/(p+1) for odd p and odd e.
From Amiram Eldar, Dec 15 2023: (Start)
Dirichlet g.f.: (1 + 1/2^(s-1)) * zeta(2*s-2) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = ((2+sqrt(2))/6) * zeta(3/2) = 1.4865345575818562471... . (End)

More terms from Matthew Conroy, Jan 19 2002

A209311 Numbers whose sum of triangular divisors is also a divisor and greater than 1.

Original entry on oeis.org

285, 1302, 1425, 1820, 2508, 3640, 3720, 4845, 4956, 5016, 5415, 7125, 7280, 9100, 9114, 9912, 11685, 12255, 12740, 14508, 15105, 16815, 17385, 18200, 19095, 19824, 20235, 20805, 22134, 22515, 23655, 23660, 24021, 24738, 25365, 25480, 27075, 27588, 27645

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			285 is in the sequence because its divisors being 1, 3, 5, 15, 19, 57, 95, 285, of which 1, 3 and 15 are triangular, these add up to 19.
1302 is in sequence because the sum of triangular divisors 21 + 6 + 3 + 1 = 31 is divisor of 1302.

Antonio Roldán, Table of n, a(n) for n = 1..13202 (terms < 10^7)

Cf. A185027, A035316, A209309, A209310.

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; fQ[n_] := Module[{tri = Total[Select[Divisors[n], TriangularQ]]}, tri > 1 && Mod[n, tri] == 0]; Select[Range[28000], fQ] (* T. D. Noe, Jan 23 2013 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^7, k=sumdiv(n, d, istriangular(d)*d); if(n/k==n\k&&k>>1, t+=1; write("b209311.txt",t," ",n)))}

A342228 Total sum of parts which are squares in all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 11, 16, 27, 42, 69, 108, 158, 229, 334, 469, 656, 903, 1255, 1685, 2283, 3032, 4033, 5290, 6936, 8986, 11650, 14969, 19172, 24402, 30998, 39110, 49260, 61712, 77155, 96000, 119209, 147394, 181958, 223713, 274533, 335792, 409980, 498981, 606273, 734572

Offset: 0

Ilya Gutkovskiy, Mar 06 2021

nonn

			For n = 4 we have:
---------------------------------
Partitions        Sum of parts
.              which are squares
---------------------------------
4 ................... 4
3 + 1 ............... 1
2 + 2 ............... 0
2 + 1 + 1 ........... 2
1 + 1 + 1 + 1 ....... 4
---------------------------------
Total .............. 11
So a(4) = 11.

Cf. A000041, A000290, A035316, A066186, A073336, A342229.

nmax = 43; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/2)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 43}]

G.f.: Sum_{k>=1} k^2*x^(k^2)/(1 - x^(k^2)) / Product_{j>=1} (1 - x^j).

a(n) = Sum_{k=1..n} A035316(k) * A000041(n-k).

A349330 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of squares (A010052).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 7, 11, 4, 2, 9, 2, 4, 4, 23, 2, 14, 2, 9, 4, 4, 2, 11, 27, 4, 12, 9, 2, 8, 2, 24, 4, 4, 4, 55, 2, 4, 4, 11, 2, 8, 2, 9, 14, 4, 2, 28, 51, 30, 4, 9, 2, 16, 4, 11, 4, 4, 2, 15, 2, 4, 14, 88, 4, 8, 2, 9, 4, 8, 2, 58, 2, 4, 30, 9, 4, 8, 2, 28, 93, 4, 2, 15, 4, 4, 4

Offset: 1

Wesley Ivan Hurt, Nov 15 2021

nonn

For each divisor d of n, add d if d is a square, otherwise add 1 [see example].

Inverse Möbius transform of n^c(n), where c = A010052. - Wesley Ivan Hurt, Mar 31 2025

			The divisors of 12 are 1, 2, 3, 4, 6, and 12 with squares 1 and 4, so a(12) = 1 + 1 + 1 + 4 + 1 + 1 = 9 (respectively).

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

Cf. A010052, A035316.

a[n_] := DivisorSum[n, If[IntegerQ @ Sqrt[#], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 15 2021 *)

a(n) = sumdiv(n, d, if (issquare(d), d, 1)); \\ Michel Marcus, Nov 15 2021

a(n) = {my(f = factor(n), cf = f, res); cf[,2]\=2; res = numdiv(f)-prod(i = 1, #f~, cf[i, 2]+1); res+=prod(i = 1, #f~, ((f[i,1]^(2*(cf[i,2]+1))-1)/(f[i,1]^2-1))); res } \\ David A. Corneth, Nov 16 2021

a(p) = 2 iff p is prime. - Wesley Ivan Hurt, Nov 28 2021

a(n) = A035316(n) + A056595(n). - R. J. Mathar, Aug 18 2024

A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 10, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, ... = n^(2/3)*A010057(n).

Winston de Greef, 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=2,k=3,n).

Cf. A035316, A069290, A333843.

gsigma := proc(n,z,k)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n,d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n,2,3),n=1..80) ;

f[p_, e_] := (p^(2*(Floor[e/3] + 1)) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)

a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ Michel Marcus, Mar 24 2023

for(n=1, 100, print1(direuler(p=2, n, (1/((1-X)*(1-p^2*X^3))))[n], ", ")) \\ Vaclav Kotesovec, Jun 26 2024

from math import prod
from sympy import factorint
def A361793(n): return prod((p**(e//3+1<<1)-1)//(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^3|n} d^2.
Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^2 * x^(k^3) / (1 - x^(k^3)). - Ilya Gutkovskiy, Jun 05 2024
From Vaclav Kotesovec, Jun 26 2024: (Start)
Dirichlet g.f.: zeta(s)*zeta(3*s-2).
Sum_{k=1..n} a(k) ~ n*(log(n) + 4*gamma - 1)/3, where gamma is the Euler-Mascheroni constant A001620. (End)

A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 9, 1, 1, 1, 73, 1, 28, 1, 9, 1, 1, 1, 9, 126, 1, 28, 9, 1, 1, 1, 73, 1, 1, 1, 252, 1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 73, 344, 126, 1, 9, 1, 28, 1, 9, 1, 1, 1, 9, 1, 1, 28, 585, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 126, 9, 1, 1, 1, 73

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 8, 0, 0, 0, 0, 27, 0, 0, ... = n^(3/2)*A010052(n).

Winston de Greef, 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=3,k=2,n).

Cf. A010052, A035316, A069290, A351308, A333843, A333844.

gsigma := proc(n,z,k)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n,d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n,3,2),n=1..80) ;

f[p_, e_] := (p^(3*(Floor[e/2] + 1)) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)

a(n) = sumdiv(n, d, if (issquare(d), sqrtint(d)^3)); \\ Michel Marcus, Mar 24 2023

from math import prod
from sympy import factorint
def A361794(n): return prod((p**(3*(e>>1)+3)-1)//(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^2|n} d^3.
Multiplicative with a(p^e) = (p^(3*(floor(e/2) + 1)) - 1)/(p^3 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^3 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024

A367989 The sum of square divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 07 2023

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

Cf. A002117, A035316, A268335, A350388, A367988.

f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 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]%2, 1, (f[i,1]^(f[i,2] + 2) - 1)/(f[i,1]^2 - 1)));}

a(n) = A035316(A350388(n)).
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) if e is even and 1 otherwise.
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2)) = 0.69451968056653021193... .

A383795 Dirichlet g.f.: zeta(2s-2) zeta(s)^2.

Original entry on oeis.org

1, 2, 2, 7, 2, 4, 2, 12, 12, 4, 2, 14, 2, 4, 4, 33, 2, 24, 2, 14, 4, 4, 2, 24, 28, 4, 22, 14, 2, 8, 2, 54, 4, 4, 4, 84, 2, 4, 4, 24, 2, 8, 2, 14, 24, 4, 2, 66, 52, 56, 4, 14, 2, 44, 4, 24, 4, 4, 2, 28, 2, 4, 24, 139, 4, 8, 2, 14, 4, 8, 2, 144, 2, 4, 56, 14, 4, 8, 2, 66, 113

Offset: 1

Vaclav Kotesovec, May 10 2025

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

Cf. A035316, A057521.

f[p_, e_] := If[OddQ[e], Sum[(e+1-2*k) * p^(2*k), {k, 0, (e-1)/2}], Sum[(e+1-2*k) * p^(2*k), {k, 0, e/2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 24 2025 *)

for(n=1, 100, print1(direuler(p=2, n, 1/((1-p^2*X^2)*(1-X)^2))[n], ", "))

Sum_{k=1..n} a(k) ~ zeta(3/2)^2 * n^(3/2)/3 - n*(log(n) + 2*log(2*Pi) + 2*gamma - 1)/2, where gamma is the Euler-Mascheroni constant A001620.

Multiplicative with a(p^e) = Sum_{k=0..(e-1)/2} (e+1-2*k) * p^(2*k) if e is odd, and Sum_{k=0..e/2} (e+1-2*k) * p^(2*k) if e is even. - Amiram Eldar, May 24 2025

cp's OEIS Frontend

A360159 a(n) is the sum of divisors of n that are odd squares.

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A055928 Sum of square divisors of n! = sum of squares of divisors of the square root of largest square dividing n!.

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A065745 Sum of squares and twice squares dividing n.

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A209311 Numbers whose sum of triangular divisors is also a divisor and greater than 1.

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A342228 Total sum of parts which are squares in all partitions of n.

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A349330 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of squares (A010052).

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A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

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A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

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A383795 Dirichlet g.f.: zeta(2s-2) zeta(s)^2.