A035316 -id:A035316 - cp's OEIS frontend

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -3, 1, 1, 1, -19, 1, 10, 1, -3, 1, 1, 1, -3, 26, 1, 10, -3, 1, 1, 1, -19, 1, 1, 1, -30, 1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -19, 50, 26, 1, -3, 1, 10, 1, -3, 1, 1, 1, -3, 1, 1, 10, -83, 1, 1, 1, -3, 1, 1, 1, -30, 1, 1, 26, -3, 1, 1, 1, -19

Offset: 1

Ilya Gutkovskiy, May 14 2021

Excess of sum of odd squares dividing n over sum of even squares dividing n.

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

Cf. A002129, A035316, A300853, A321543, A344299.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, IntegerQ[#^(1/2)] &], {n, 1, 80}]
f[p_, e_] := (p^(2*Floor[e/2] + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*Floor[e/2] + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*d)); \\ Michel Marcus, Aug 22 2021

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 2 - (2^(2*floor(f[i,2]/2) + 2) - 1)/3, (f[i,1]^(2*floor(f[i,2]/2) + 2) - 1)/(f[i,1]^2 - 1)));} \\ Amiram Eldar, Nov 15 2022

Multiplicative with a(2^e) = 2 - (2^(2*floor(e/2) + 2) - 1)/3, and a(p^e) = (p^(2*floor(e/2) + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 15 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

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 11, 16, 26, 27, 28, 33, 34, 35, 36, 57, 58, 68, 69, 74, 75, 76, 77, 82, 108, 109, 119, 124, 125, 126, 127, 148, 149, 150, 151, 201, 202, 203, 204, 209, 210, 211, 212, 217, 227, 228, 229, 250, 300, 326, 327, 332, 333, 343, 344, 349, 350, 351, 352, 357, 358, 359, 369, 454, 455, 456

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

nonn

Partial sums of A035316.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..10000

Cf. A013936, A024916, A035316, A309126, A309127.

Table[Sum[k^2 Floor[n/k^2], {k, 1, n}], {n, 1, 66}]
nmax = 66; CoefficientList[Series[1/(1 - x) Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, k^2*(n\k^2)); \\ Seiichi Manyama, Aug 30 2021

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

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

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.

A300853 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, 3, 1, -1, 1, -5, 10, -1, 1, 3, 1, -1, 1, 11, 1, -10, 1, 3, 1, -1, 1, -5, 26, -1, 10, 3, 1, -1, 1, -21, 1, -1, 1, 30, 1, -1, 1, -5, 1, -1, 1, 3, 10, -1, 1, 11, 50, -26, 1, 3, 1, -10, 1, -5, 1, -1, 1, 3, 1, -1, 10, 43, 1, -1, 1, 3, 1, -1, 1, -50, 1, -1, 26, 3, 1, -1, 1, 11, 91, -1, 1, 3, 1

Offset: 1

Ilya Gutkovskiy, Mar 13 2018

			L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 - x^6/6 + x^7/7 - 5*x^8/8 + 10*x^9/9 - x^10/10 + ...
exp(L(x)) = 1 + x + x^4 + x^5 + x^9 + x^10 + x^13 + x^14 + ... + A033461(n)*x^n + ...

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000290, A033461, A035316, A039956, A056911, A304876, A304906.

Cf. A078434, A268682.

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2), {k, 1, Floor[nmax^(1/2) + 1]}]], {x, 0, nmax}],x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 + x^k^2), {k, 1,Floor[nmax^(1/2) + 1]}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/2)] &], {n, 85}]
f[p_, e_] := If[p == 2, (1 - (-2)^(e + 1))/3, (p^(2*Floor[e/2 + 1]) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

seq(n)={Vec(deriv(log(prod(k=1, n, (1 + x^(k^2) + O(x*x^n))))))} \\ Andrew Howroyd, Jul 20 2018

a(n)={sumdiv(n, d, if(n%d^2, 0, (-1)^(n/d^2 + 1) * d^2))} \\ Andrew Howroyd, Jul 20 2018

G.f.: Sum_{k>=1} k^2*x^(k^2)/(1 + x^(k^2)).
a(n) = 1 if n is an odd squarefree (A056911).
a(n) = -1 if n is an even squarefree (A039956).
a(n) = Sum_{d^2|n} (-1)^(n/d^2 + 1) * d^2. - Andrew Howroyd, Jul 20 2018
Multiplicative with a(2^e) = (1 - (-2)^(e + 1))/3, and a(p^e) = (p^(2*floor(e/2 + 1)) - 1)/(p^2 - 1) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1 - 1/sqrt(2)) * zeta(3/2)/3 = A268682 * A078434 / 3 = 0.255049... . (End)

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A359937 a(n) = Sum_{d|n, d-1 is square} d.

Original entry on oeis.org

1, 3, 1, 3, 6, 3, 1, 3, 1, 18, 1, 3, 1, 3, 6, 3, 18, 3, 1, 18, 1, 3, 1, 3, 6, 29, 1, 3, 1, 18, 1, 3, 1, 20, 6, 3, 38, 3, 1, 18, 1, 3, 1, 3, 6, 3, 1, 3, 1, 68, 18, 29, 1, 3, 6, 3, 1, 3, 1, 18, 1, 3, 1, 3, 71, 3, 1, 20, 1, 18, 1, 3, 1, 40, 6, 3, 1, 29, 1, 18, 1, 85, 1, 3

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

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

Cf. A002522, A035316, A359967.

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)); Vec(sum(k=0, sqrtint(N), (k^2+1)*x^(k^2+1)/(1-x^(k^2+1))))

G.f.: Sum_{k>=0} (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

A232554 Square numbers whose sum of square divisors is also square.

Original entry on oeis.org

1, 1764, 60516, 82369, 529984, 2056356, 2798929, 3534400, 18181696, 38900169, 96020401, 97121025, 335988900, 455907904, 457318225, 617820736, 1334513961, 1599200100, 2176689025, 3279852900, 4464244225, 8586616896, 15688815025, 24514164900, 33366502225

Offset: 1

Antonio Roldán, Nov 26 2013

nonn

			60516 = 246^2. Sum of square divisors: 60516 + 15129 + 6724 + 1681 + 36 + 9 + 4 + 1 = 84100 = 290^2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A035316, A046655, A232555, A232556, A232557.

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); A035316[1] = 1; A035316[n_] := Times @@ f @@@ FactorInteger[n];
Select[Range[200000]^2, IntegerQ[Sqrt[A035316[#]]]&] (* Amiram Eldar, Aug 12 2023 *)
ssdQ[n_]:=IntegerQ[Sqrt[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]]]]; Select[Range[200000]^2,ssdQ] (* Harvey P. Dale, Feb 03 2025 *)

{for(n=1,10^5,m=n*n;k=sumdiv(m,d,d*issquare(d));if(issquare(k),print(m)))}

a(n) = A046655(n)^2.

A232555 Nonsquare numbers whose sum of proper square divisors is a square greater than 1.

Original entry on oeis.org

3528, 5292, 8820, 10584, 12348, 17640, 19404, 22932, 24696, 26460, 29988, 33516, 37044, 38808, 40572, 45864, 51156, 52920, 54684, 58212, 59976, 61740, 65268, 67032, 68796, 72324, 74088, 75852, 81144, 82908, 89964, 93492, 97020, 100548, 102312, 104076, 107604

Offset: 1

Antonio Roldán, Nov 26 2013

nonn

			8820 is nonsquare number. Sum of proper square divisor: 1764 + 441 + 196 + 49 + 36 + 9 + 4 + 1 = 2500 = 50^2.

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

Subsequence of A232556.

Cf. A232554, A232557.

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); A035316[1] = 1; A035316[n_] := Times @@ f @@@ FactorInteger[n]; sqQ[n_] := n>1 && IntegerQ[Sqrt[n]];
Select[Range[100000], !IntegerQ[Sqrt[#]] && sqQ[A035316[#]] &] (* Amiram Eldar, Aug 12 2023 *)

{for(n=1,10^5,if(issquare(n)==0,k=sumdiv(n,d,d*issquare(d)); if(issquare(k)&&k>>1, print(n))))}

A232556 Numbers whose sum of proper square divisors is a square greater than 1.

Original entry on oeis.org

900, 3528, 4900, 5292, 8820, 10404, 10584, 12348, 17640, 19404, 22932, 24696, 26460, 29988, 33516, 37044, 38808, 40572, 45864, 51156, 52920, 54684, 58212, 59976, 61740, 65268, 67032, 68796, 72324, 74088, 75852, 79524, 81144, 81796, 82908, 89964, 93492

Offset: 1

Antonio Roldán, Nov 26 2013

nonn

			Sum of proper square divisors of 5292: 1764 + 441 + 196 + 49 + 36 + 9 + 4 + 1 = 2500 = 50^2 is square number.

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

Disjoint union of A232555 and A232557.

Cf. A232554.

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); A035316[1] = 1; A035316[n_] := Times @@ f @@@ FactorInteger[n]; sqQ[n_] := n>1 && IntegerQ[Sqrt[n]];
Select[Range[100000], sqQ[A035316[#] - If[IntegerQ[Sqrt[#]], #, 0]] &] (* Amiram Eldar, Aug 12 2023 *)

{for(n=1,10^5,k=sumdiv(n,d,d*issquare(d)*(d>1,print(n)))}

cp's OEIS Frontend

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385006 The sum of the biquadratefree divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A300853 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))) = Sum_{n>=1} a(n)*x^n/n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A359937 a(n) = Sum_{d|n, d-1 is square} d.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI