A046655 -id:A046655 - cp's OEIS frontend

A046656 Square roots of sums of squares of divisors in A046655.

1, 50, 290, 290, 850, 1690, 1690, 2210, 4930, 6710, 9802, 10660, 22100, 24650, 22100, 28730, 38918, 48100, 48100, 68900, 68900, 114070, 128180, 188500, 188500, 246500, 226798, 246500, 278980, 404260, 399620, 490100, 640900, 640900, 817700, 746980, 906100

Offset: 1

Labos Elemer

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..500 from Donovan Johnson)

Cf. A001157, A046655.

Select[Sqrt[DivisorSigma[2, #]] & /@ Range[750000], IntegerQ] (* Jayanta Basu, Jun 27 2013 *)

a(n) = sqrt(A001157(A046655(n))). - Amiram Eldar, Aug 12 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.

A290846 Numbers k such that sum of squares of divisors of k is a cube.

Original entry on oeis.org

1, 4182, 4879, 122664360, 315870765, 4621320120, 5006430130, 20245909880, 220899101736, 239307360214, 812593845485

Offset: 1

Altug Alkan, Aug 12 2017

It is known that A006532 is an infinite sequence. Is this sequence infinite?

a(12) > 10^12. - Giovanni Resta, Aug 14 2017

			4879 is a term because 1^2 + 7^2 + 17^2 + 41^2 + 119^2 + 287^2 + 697^2 + 4879^2 = 290^3.

Giovanni Resta, A list of 3110 terms > 10^12

Cf. A001157, A006532, A046655.

PARI
```
isok(n) = ispower(sigma(n, 2), 3);
```

a(6)-a(11) from Giovanni Resta, Aug 14 2017

A063831 Sum of cubes of divisors is a square.

Original entry on oeis.org

1, 2, 345, 690, 47196, 46248900, 53262468, 71315748, 140553735, 188194335, 215515727, 281107470, 288564647, 292978595, 310129096, 376388670, 431031454, 577129294, 585957190, 1474108335, 2133051720, 2489605188, 2948216670, 3270679304, 4043104611, 5142743032

Offset: 1

Jason Earls, Aug 21 2001

nonn

			345 is in the sequence since sigma_3(345) = 6552^2 is a square.

Cf. A001158, A006532, A046655.

Select[Range[10^5], IntegerQ[ Sqrt[ DivisorSigma[3,#] ] ] &] (* Amiram Eldar, Dec 06 2018 *)

for(n=1,10^5, if(issquare(sigma(n,3)),print1(n, ", ")))

More terms from Thomas Baruchel, Oct 20 2003
More terms from Sean A. Irvine, Sep 24 2009
a(24)-a(26) from Amiram Eldar, Dec 06 2018

A318169 Composite numbers k such that sigma_2(k) - 1 is a square, where sigma_2(k) = A001157(k) is the sum of squares of divisors of k.

Original entry on oeis.org

6, 40, 136, 2696, 3352, 46976, 223736, 5509736, 1915798072

Offset: 1

Amiram Eldar, Aug 20 2018

This property is shared with all the primes since sigma_2(p) = 1 + p^2.
The values of sqrt(sigma_2(a(n))-1) are 7, 47, 157, 3107, 3863, 54243, 257843, 6349657, 2207848187.
Are there terms not of the form 2^k * p where p is prime? - David A. Corneth, Aug 20 2018
2*10^12 < a(10) <= 44463118771144. The terms 21687324345660824, 14524130539077100050485512, 287674439504279743204606472 (and others) of the form 2^k * p can be found by solving the quadratic Diophantine equation sigma_2(2^k) * (p^2 + 1) = x^2 + 1 for appropriate values of k. - Giovanni Resta, Aug 20 2018

Cf. A001157, A046655, A289290.

[n: n in [2..6*10^6] |not IsPrime(n) and IsSquare(DivisorSigma(2, n)-1)]; // Vincenzo Librandi, Aug 22 2018

sQ[n_] := IntegerQ[Sqrt[n]]; aQ[n_] := CompositeQ[n] && sQ[DivisorSigma[2,n]-1]; Select[Range[10000],aQ]

forcomposite(n=2, 1e15, if( issquare(sigma(n,2)-1), print1(n, ", ")))

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A046656 Square roots of sums of squares of divisors in A046655.

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

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A290846 Numbers k such that sum of squares of divisors of k is a cube.

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A063831 Sum of cubes of divisors is a square.

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A318169 Composite numbers k such that sigma_2(k) - 1 is a square, where sigma_2(k) = A001157(k) is the sum of squares of divisors of k.

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