A035316 -id:A035316 - cp's OEIS frontend

A351316 Sum of the 10th powers of the square divisors of n.

1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1, 1048577, 1, 1, 1, 1099512676353, 1, 3486784402, 1, 1048577, 1, 1, 1, 1048577, 95367431640626, 1, 3486784402, 1048577, 1, 1, 1, 1099512676353, 1, 1, 1, 3656161927895954, 1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^10 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 1099512676353; a(16) = Sum_{d^2|16} (d^2)^10 = (1^2)^10 + (2^2)^10 + (4^2)^10 = 1099512676353.

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

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (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), this sequence (k=10).

Cf. A010052, A008836, A013958.

f[p_, e_] := (p^(20*(1 + Floor[e/2])) - 1)/(p^20 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^10],{n,50}] (* Harvey P. Dale, Aug 24 2024 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^20*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

a(n) = Sum_{d^2|n} (d^2)^10.
Multiplicative with a(p) = (p^(20*(1+floor(e/2))) - 1)/(p^20 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^20*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-20).
Sum_{k=1..n} a(k) ~ (zeta(21/2)/21) * n^(21/2). (End)
a(n) = Sum_{d|n} d^10 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^10*sigma_10(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A359967 a(n) = Sum_{d|n, d+1 is square} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 8, 3, 0, 0, 3, 0, 0, 18, 8, 0, 3, 0, 0, 3, 0, 0, 35, 0, 0, 3, 0, 0, 18, 0, 8, 3, 0, 35, 3, 0, 0, 3, 8, 0, 3, 0, 0, 18, 0, 0, 83, 0, 0, 3, 0, 0, 3, 0, 8, 3, 0, 0, 18, 0, 0, 66, 8, 0, 3, 0, 0, 3, 35, 0, 35, 0, 0, 18, 0, 0, 3, 0, 88, 3, 0, 0, 3, 0, 0

Offset: 1

Seiichi Manyama, Jan 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^7

Cf. A005563, A035316, A179952, A359937.

Table[Sum[If[IntegerQ[Sqrt[d+1]], d, 0], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 21 2023 *)

PARI
```
a(n) = sumdiv(n, d, issquare(d+1)*d);
```

my(N=100, x='x+O('x^N)); concat([0, 0], Vec(sum(k=2, sqrtint(N+1), (k^2-1)*x^(k^2-1)/(1-x^(k^2-1)))))

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

Sum_{k=1..n} a(k) ~ zeta(3/2)*n^(3/2)/3. - Vaclav Kotesovec, Jan 21 2023

A365172 The sum of divisors d of n such that gcd(d, n/d) is a square.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 21, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 45, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 84, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68

Offset: 1

Amiram Eldar, Aug 25 2023

The number of these divisors is A365171(n).

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

Cf. A000203, A005117, A034448, A035316, A046101, A365171, A365174.

f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), (1 + p^(2*Floor[(e + 1)/4] + 1))*(p^(2*Floor[e/4] + 2) - 1)/(p^2 - 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]; if(e%2, (1 + p^(2*((e+1)\4)+1))*(p^(2*(e\4)+2) - 1)/(p^2 - 1), (p^(e+2) - 1)/(p^2 - 1)));}

Multiplicative with a(p^e) = (p^(e+2) - 1)/(p^2 - 1) if e is even, and (1 + p^(2*floor((e+1)/4) + 1))*(p^(2*floor(e/4)+2) - 1)/(p^2 - 1) if e is odd.
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034448(n), with equality if and only if n is not a biquadrateful number (A046101).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/(2 * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^5)) = 0.696082796052... .

A373439 Numerator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

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, 25, 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, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

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

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{d^2|n} 1/d^2.
From Amiram Eldar, Jun 26 2024: (Start)
Let f(n) = a(n)/A373440(n). Then:
f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).
Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).
Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)

A373440 Denominator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

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

Cf. A007407, A007947, A017666, A017668, A035316, A332881, A373439 (numerators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Denominator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = denominator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Denominators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1-x^(k^2))).

a(n) is the denominator of Sum_{d^2|n} 1/d^2.

A380324 The sum of the squares dividing the n-th exponentially odd number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 20 2025

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

Cf. A035316, A268335, A380323, A380325.

f[p_, e_] := If[OddQ[e], (p^(e+1) - 1)/(p^2 - 1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1]^2 - 1), 0));}
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

a(n) = A035316(A268335(n)).

A071326 Sum of squares > 1 dividing n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 0, 0, 20, 0, 9, 0, 4, 0, 0, 0, 4, 25, 0, 9, 4, 0, 0, 0, 20, 0, 0, 0, 49, 0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 20, 49, 25, 0, 4, 0, 9, 0, 4, 0, 0, 0, 4, 0, 0, 9, 84, 0, 0, 0, 4, 0, 0, 0, 49, 0, 0, 25, 4, 0, 0, 0, 20, 90, 0, 0, 4, 0, 0, 0, 4, 0, 9, 0, 4, 0, 0, 0, 20, 0, 49, 9, 129

Offset: 1

Reinhard Zumkeller, May 18 2002

nonn

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

Cf. A000203, A010052, A071325, A071327.

One less than A035316.

Array[DivisorSum[#, # &, IntegerQ@ Sqrt@ # &] - 1 &, 100] (* Michael De Vlieger, Nov 17 2017 *)
f[p_, e_] := (p^(2*(1+Floor[e/2]))-1)/(p^2-1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* Amiram Eldar, May 15 2025 *)

A071326(n) = sumdiv(n,d,(d>1)*issquare(d)*d); \\ Antti Karttunen, Nov 17 2017

a(n) = A035316(n)-1.

Sum_{d|n, d>1} A010052(d). - Antti Karttunen, Nov 17 2017

More terms from Antti Karttunen, Nov 17 2017

A276559 Expansion of Sum_{k>=1} k^2x^k^2/(1 - x^k^2) Product_{k>=1} 1/(1 - x^k^2).

Original entry on oeis.org

1, 2, 3, 8, 10, 12, 14, 24, 36, 40, 44, 60, 78, 84, 90, 128, 153, 180, 190, 240, 273, 308, 322, 384, 475, 520, 567, 644, 754, 810, 868, 992, 1122, 1258, 1330, 1548, 1702, 1862, 1950, 2200, 2460, 2646, 2838, 3124, 3510, 3726, 3948, 4320, 4802, 5200, 5457, 6032, 6572, 7128, 7425, 8064, 8778, 9454, 9971, 10680

Offset: 1

Ilya Gutkovskiy, Apr 10 2017

nonn

Sum of all parts of all partitions of n into squares.

Convolution of the sequences A001156 and A035316.

			a(8) = 24 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 3*8 = 24.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for related partition-counting sequences

Cf. A000290, A001156, A035316, A066186, A243148, A281541.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->
      `if`(s>n, 0, (p->p+[0, p[1]*s])(b(n-s, i))))(i^2)+b(n, i-1))
    end:
a:= n-> b(n, isqrt(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 19 2018

nmax = 60; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 - x^k^2), {k, 1, nmax}] Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 60; Rest[CoefficientList[Series[x D[Product[1/(1 - x^k^2), {k, 1, nmax}], x], {x, 0, nmax}], x]]

G.f.: Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).
G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k^2).
a(n) = n * A001156(n).
a(n) = n * Sum_{k=1..n} A243148(n,k). - Alois P. Heinz, Sep 19 2018

A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2k^2)) / (1 - x^(2k^2))^2.

Original entry on oeis.org

1, 0, 3, 1, 5, 0, 7, 0, 10, 0, 11, 3, 13, 0, 15, 1, 17, 0, 19, 5, 21, 0, 23, 0, 26, 0, 30, 7, 29, 0, 31, 0, 33, 0, 35, 10, 37, 0, 39, 0, 41, 0, 43, 11, 50, 0, 47, 3, 50, 0, 51, 13, 53, 0, 55, 0, 57, 0, 59, 15, 61, 0, 70, 1, 65, 0, 67, 17, 69, 0, 71, 0, 73, 0, 78, 19, 77, 0, 79, 5, 91

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

Sum of odd divisors d of n such that n/d is square.

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

Cf. A000593, A010052, A035316, A036554 (positions of 0's), A056911 (fixed points), A076752, A193356, A328372.

a:=[];for n in [1..81] do  v:=[d:d in Divisors(n)| IsOdd(d) and IsSquare(n div d)]; if #v ne 0  then Append(~a,&+v); else Append(~a,0); end if; end for; a; // Marius A. Burtea, Oct 14 2019

nmax = 81; CoefficientList[Series[Sum[x^(k^2) (1 + x^(2 k^2))/(1 - x^(2 k^2))^2, {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 81}]
f[p_, e_] := If[p == 2, Boole @ EvenQ[e], If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), (p^(e + 2) - p)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 16 2020 *)

a(n) = sumdiv(n, d, if ((d%2) && issquare(n/d), d)); \\ Michel Marcus, Oct 14 2019

G.f.: Sum_{k>=1} (2*k - 1) * (theta_3(x^(2*k - 1)) - 1) / 2.
G.f.: Sum_{i>=1} Sum_{j>=1} phi(i) * x^(i*j^2) / (1 + x^(i*j^2)).
Dirichlet g.f.: (1 - 2^(1 - s)) * zeta(s-1) * zeta(2*s).
a(n) = Sum_{d|n} A193356(d) * A010052(n/d).
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 360. - Vaclav Kotesovec, Oct 14 2019
Multiplicative with a(2^e) = 0 if e is odd, and 1 if e is even, and for p > 2, a(p^e) = (p^(e + 2) - p)/(p^2 - 1) if e is odd, and (p^(e + 2) - 1)/(p^2 - 1) if e is even. - Amiram Eldar, Oct 16 2020

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

A351316 Sum of the 10th powers of the square divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A359967 a(n) = Sum_{d|n, d+1 is square} d.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A365172 The sum of divisors d of n such that gcd(d, n/d) is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A373439 Numerator of sum of reciprocals of square divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A373440 Denominator of sum of reciprocals of square divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A380324 The sum of the squares dividing the n-th exponentially odd number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A071326 Sum of squares > 1 dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A276559 Expansion of Sum_{k>=1} k^2*x^k^2/(1 - x^k^2) * Product_{k>=1} 1/(1 - x^k^2).

Views

A276559 Expansion of Sum_{k>=1} k^2x^k^2/(1 - x^k^2) Product_{k>=1} 1/(1 - x^k^2).

A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2k^2)) / (1 - x^(2k^2))^2.