A074386 -id:A074386 - cp's OEIS frontend

A008848 Squares whose sum of divisors is a square.

1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041

Offset: 1

N. J. A. Sloane

nonn

Solutions to sigma(x^2) = (2k+1)^2. - Labos Elemer, Aug 22 2002
Intersection of A006532 and A000290. The product of any two coprime terms is also in this sequence. - Charles R Greathouse IV, May 10 2011
Also intersection of A069070 and A000290. - Michel Marcus, Oct 06 2013
Conjectures: (1) a(2) = 81 is the only prime power (A246655) in this sequence. (2) 81 and 400 are only terms x for which sigma(x) is in A246655. (3) x = 1 is the only such term that sigma(x) is also a term. See also comments in A074386, A336547 and A350072. - Antti Karttunen, Jul 03 2023, (2) corrected in May 11 2024

			n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=400: sigma(400) = sigma(16)*sigma(25) = 31*31 = 961.
n=32400 (= 81*400): sigma(32400) = 116281 = 341^2 = 121*961.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..400

Terms of A008847 squared.
Cf. A028982, A001248, A000203, A074386, A231484 (odd terms), A246655, A336547, A350072.
Subsequence of A000290, of A006532, and of A069070.

Do[s=DivisorSigma[1, n^2]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[n^2]], {n, 1, 10000000}] (* Labos Elemer *)
Select[Range[320000]^2,IntegerQ[Sqrt[DivisorSigma[1,#]]]&] (* Harvey P. Dale, Feb 22 2015 *)

for(n=1,1e6,if(issquare(sigma(n^2)), print1(n^2", "))) \\ Charles R Greathouse IV, May 10 2011

a(n) = A008847(n)^2.

A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

Original entry on oeis.org

1, 1, 3, 1, 7, 2, 1, 5, 2, 3, 1, 4, 2, 3, 8, 1, 7, 2, 3, 2401, 5, 1, 29, 2, 3, 6, 5, 4, 1, 10, 2, 3, 9, 5, 4, 7, 1, 19, 2, 3, 13, 5, 4, 7, 10, 1, 6, 2, 3, 8, 5, 4, 7, 18, 19, 1, 9, 2, 3, 24, 5, 4, 7, 16, 21, 43, 1, 13, 2, 3, 2401, 5, 4, 7, 49, 31213, 9604, 6, 1, 8, 2, 3, 10, 5, 4, 7, 33, 22

Offset: 1

Labos Elemer, Aug 26 2002

In the table output, one can observe constant diagonals (or lines in the square output). The indices of these are: 1, 3, 4, 6, 7, 8, 12, 13, ... (see A002191). And the corresponding values are: 1, 2, 3, 5, 4, 7, 6, 9, ... (see A002192). - Michel Marcus, Dec 19 2013

			Triangle begins
1;
1,3;
1,7,2;
1,5,2,3;
1,4,2,3,8; ...

Cf. A000203, A028982, A074384, A074386, A072461, A072462, A072862.

{k=0, s=0, fl=1}; Table[Print["#"]; Table[fl=1; Print[{r, m}]; Do[s=Mod[DivisorSigma[1, n], m]; If[(s==r)&&(fl==1), Print[n]; fl=0], {n, 1, 150000}], {r, 0, m-1}], {m, 1, 25}]

Min{x; Mod[sigma[x], n]=r}, r=1..n, n=1, ...

A074388 Numbers of the form 2k^2 such that sigma(2k^2) is an odd square.

Original entry on oeis.org

195938, 224450, 13645088, 15870978, 18180450, 29184800, 1105252128, 2363968800, 2686005218, 2917410498, 3564550178, 5056357922, 8442721568, 10814792450, 18587462432, 21292224800, 48666384162, 140836104992, 212352534818, 217566422658, 288728564418, 315325993248

Offset: 1

Labos Elemer, Aug 22 2002

nonn

No terms whose sum of divisors is a square of a prime below 10^12 were found.

			195938 = 2*313^2 and sigma(195938) = 294849 = 543^2.

Amiram Eldar, Table of n, a(n) for n = 1..200 (calculated using the b-file at A097023)

Cf. A000203, A001248, A028982, A074386, A097023.

Do[s=DivisorSigma[1, 2*(n^2)]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[2*(n^2)]], {n, 1, 10000000}]

a(n) = 2*A097023(n)^2. - Amiram Eldar, Aug 13 2024

a(19)-a(22) from Amiram Eldar, Aug 13 2024

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A008848 Squares whose sum of divisors is a square.

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A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

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A074388 Numbers of the form 2k^2 such that sigma(2k^2) is an odd square.

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cp's OEIS Frontend

A008848 Squares whose sum of divisors is a square.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

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Author

Keywords

Comments

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Programs

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Formula

A074388 Numbers of the form 2*k^2 such that sigma(2*k^2) is an odd square.

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A074388 Numbers of the form 2k^2 such that sigma(2k^2) is an odd square.