A035316 -id:A035316 - cp's OEIS frontend

A232557 Square numbers whose sum of proper square divisors is also square greater than 1.

900, 4900, 10404, 79524, 81796, 417316, 532900, 846400, 1542564, 2464900, 3232804, 3334276, 3496900, 12432676, 43850884, 50836900, 51811204, 71470116, 107453956, 236975236, 253892356, 432889636, 544102276, 864948100, 1192597156, 1450543396

Offset: 1

Antonio Roldán, Nov 26 2013

nonn

			10404 = 102^2 is a square number. Sum of proper square divisor of 10404 is 2601 + 1156 + 289 + 36 + 9 + 4 + 1 = 4096 = 64^2.

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

Subsequence of A232556.

Cf. A035316, A232554, A232555.

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[40000]^2, sqQ[A035316[#] - #]&] (* Amiram Eldar, Aug 12 2023 *)

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

A285309 Sum of nonsquare divisors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 23, 13, 23, 23, 10, 17, 29, 19, 37, 31, 35, 23, 55, 5, 41, 30, 51, 29, 71, 31, 42, 47, 53, 47, 41, 37, 59, 55, 85, 41, 95, 43, 79, 68, 71, 47, 103, 7, 67, 71, 93, 53, 110, 71, 115, 79, 89, 59, 163, 61, 95, 94, 42, 83, 143, 67, 121, 95, 143, 71, 145, 73, 113, 98

Offset: 1

Ilya Gutkovskiy, Apr 16 2017

nonn

			a(6) = 11 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are nonsquares {2, 3, 6} therefore 2 + 3 + 6 = 11.

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

Cf. A000037, A000203, A005117, A035316, A056595, A087153.

Table[DivisorSum[n, # &, Mod[DivisorSigma[0, #], 2] == 0 &], {n, 1, 75}]
nmax = 75; Rest[CoefficientList[Series[Sum[(k + Floor[1/2 + Sqrt[k]]) x^(k + Floor[1/2 + Sqrt[k]])/(1 - x^(k + Floor[1/2 + Sqrt[k]])), {k, 1, nmax}], {x, 0, nmax}], x]]
Array[DivisorSum[#, # &, ! IntegerQ@ Sqrt@ # &] &, 75] (* Michael De Vlieger, Nov 23 2017 *)

a(n) = sumdiv(n, d, if (!issquare(d), d)); \\ Michel Marcus, Apr 17 2017

import gmpy
from sympy import divisors
def a(n): return sum([d for d in divisors(n) if gmpy.is_square(d)==0]) # Indranil Ghosh, Apr 18 2017

G.f.: Sum_{k>=1} A000037(k)*x^A000037(k)/(1 - x^A000037(k)).
a(n) = A000203(n) - A035316(n).
a(A005117(n)) = A000203(A005117(n)) - 1.
a(p^(2*k-1)) = a(p^(2*k)) = p*(p^(2*k) - 1)/(p^2 - 1) for p is a prime and k >= 1.

A293235 a(n) is the sum of proper divisors of n that are square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 14, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 1, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 21, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 21, 1, 50, 10, 30, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - Robert Israel, Oct 08 2017

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

Cf. A010052, A035316, A078434, A293228, A293234.

A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
f:= n -> A035316(n) - `if`(issqr(n),n,0):
map(f, [$1..100]); # Robert Israel, Oct 08 2017

Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* Harvey P. Dale, Dec 29 2023 *)

PARI
```
A293235(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA010052(d)*d.
a(n) = A035316(n) - (A010052(n)*n).
G.f.: Sum_{k>=1} k^2 * x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - Amiram Eldar, Dec 01 2023

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

A326058 a(n) = n - {the sum of square divisors of n}.

Original entry on oeis.org

0, 1, 2, -1, 4, 5, 6, 3, -1, 9, 10, 7, 12, 13, 14, -5, 16, 8, 18, 15, 20, 21, 22, 19, -1, 25, 17, 23, 28, 29, 30, 11, 32, 33, 34, -14, 36, 37, 38, 35, 40, 41, 42, 39, 35, 45, 46, 27, -1, 24, 50, 47, 52, 44, 54, 51, 56, 57, 58, 55, 60, 61, 53, -21, 64, 65, 66, 63, 68, 69, 70, 22, 72, 73, 49, 71, 76, 77, 78, 59, -10, 81, 82, 79

Offset: 1

Antti Karttunen, Jun 06 2019

sign

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

Cf. A033879, A035316, A326059, A326060.

A035316(n) = sumdiv(n,d,issquare(d)*d);
A326058(n) = (n-A035316(n));

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

a(n) = A033879(n) + A326059(n).

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)).

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

A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.

Original entry on oeis.org

6, 12, 18, 24, 48, 54, 96, 102, 110, 114, 138, 162, 174, 186, 192, 204, 220, 222, 228, 246, 258, 282, 315, 318, 348, 354, 364, 366, 372, 384, 402, 414, 426, 438, 440, 444, 456, 474, 486, 492, 498, 516, 522, 534, 550, 558, 564, 582, 606, 618, 636, 642, 654, 678

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			186 is a term because the sum of its triangular divisors, 1+3+6 = 10 is also triangular.

Antonio Roldán, Table of n, a(n) for n = 1..8055 (terms <= 10^5)

Cf. A000217, A035316, A185027, A209310.

triQ[n_] := n > 1 && IntegerQ[Sqrt[8*n+1]]; q[n_] := triQ[1 + DivisorSum[n, #&, triQ[#] &]]; Select[Range[700], q] (* Amiram Eldar, Aug 12 2023 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^5, k=sumdiv(n, d, istriangular(d)*d); if(istriangular(k)&&k>>1, t+=1; write("b209309.txt",t," ",n)))}

A209310 Triangular numbers whose sum of triangular divisors is also triangular and greater than 1.

Original entry on oeis.org

6, 4186, 32131, 52975, 78210, 111628, 237016, 247456, 584821, 750925, 1464616, 3649051, 5791906, 11297881, 16082956, 24650731, 27243271, 38618866, 46585378, 51546781, 56026405, 76923406, 89880528, 96070591, 126906346, 164629585, 201854278, 228733966

Offset: 1

Antonio Roldán, Jan 18 2013

nonn

			4186 is in sequence because it is triangular (4186 = 91*92/2) and the sum of its triangular divisors, 4186+91+1 = 4278 is also triangular (4278 = 92*93/2).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Subsequence of A209309.

Cf. A000217, A185027, A035316.

triQ[n_] := n > 1 && IntegerQ[Sqrt[8*n+1]]; q[n_] := triQ[1 + DivisorSum[n, #&, triQ[#] &]]; Select[Accumulate[Range[22000]], q] (* Amiram Eldar, Aug 12 2023 *)

istriangular(n)=issquare(8*n+1)
{t=0; for(n=1, 10^8, if(istriangular(n), k=sumdiv(n, d, istriangular(d)*d) ;if(istriangular(k)&&k>>1,t+=1;write("b209310.txt",t," ",n))))}

A225882 Numbers k such that core(k) is equal to the sum of the proper square divisors of k, where core(k) = A007913(k).

Original entry on oeis.org

20, 90, 336, 650, 5440, 7371, 13000, 14762, 28730, 30240, 83810, 87296, 130682, 147420, 218400, 280370, 295240, 406875, 708122, 924482, 1397760, 1875530, 2613640, 3536000, 4881890, 4960032, 5884851, 7856640, 7893290, 8137500

Offset: 1

Antonio Roldán, May 19 2013

nonn

If p is prime and p^2 + 1 squarefree, then p^2*(p^2 + 1) is in the sequence.

			13000 is a term because core(13000) = 130 = 100 + 25 + 4 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..282 (terms < 5*10^11; first 80 terms from Charles R Greathouse IV)

Cf. A007913, A035316, A225880, A225881.

for(n=2,10^8,if(core(n)==sumdiv(n,d,d*issquare(d)),print(n)))

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

cp's OEIS Frontend

A232557 Square numbers whose sum of proper square divisors is also square greater than 1.

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A285309 Sum of nonsquare divisors of n.

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A293235 a(n) is the sum of proper divisors of n that are square.

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A300909 Sum of 4th powers dividing n.

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A326058 a(n) = n - {the sum of square divisors of n}.

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A385005 The sum of the cubefull divisors of n.

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

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A209309 Numbers whose sum of triangular divisors is also triangular and greater than 1.