A008847 -id:A008847 - cp's OEIS frontend

A163763 Sqrt(sigma(A008847(n)^2)), where A008847 lists m such that sigma(m^2) is a square.

1, 11, 31, 341, 1729, 2821, 19019, 31031, 43617, 43491, 68961, 72219, 82677, 86583, 117831, 117831, 187131, 347529, 347529, 479787, 503347, 414309, 436107, 496713, 478401, 503347, 758571, 794409, 909447, 952413, 658749, 696787, 696787

Offset: 1

M. F. Hasler, Aug 03 2009

nonn

Note that a(k)=a(k+1) for k=15, 18, 32, 90, 231,... - Zak Seidov, May 02 2016

Zak Seidov and Donovan Johnson, Table of n, a(n) for n = 1..400 (first 161 terms from Zak Seidov)

Cf. A163764.

Select[Sqrt@ DivisorSigma[1, #^2] &@ Range[10^6], IntegerQ] (* Michael De Vlieger, May 02 2016, after Harvey P. Dale at A008847 *)

lista(nn) = {for (n=1, nn, if (issquare(s = sigma(n^2)), print1(sqrtint(s), ", ")););} \\ Michel Marcus, May 02 2016

A008848 Squares whose sum of divisors is a square.

Original entry on oeis.org

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.

A097023 Numbers k such that the sum of the divisors of 2*k^2 is a square.

Original entry on oeis.org

313, 335, 2612, 2817, 3015, 3820, 23508, 34380, 36647, 38193, 42217, 50281, 64972, 73535, 96404, 103180, 155991, 265364, 325847, 329823, 379953, 397068, 452529, 476545, 584748, 624748, 661815, 668660, 867636, 928620

Offset: 1

Labos Elemer, Aug 24 2004

nonn

			sigma(2*313^2) = 543^2.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000203, A008847, A074388, A097024.

Select[Range[950000],IntegerQ[Sqrt[DivisorSigma[1,2#^2]]]&] (* Harvey P. Dale, Jul 10 2012 *)

a(n) = sqrt(A074388(n)/2). - Amiram Eldar, Aug 13 2024

Definition clarified by Harvey P. Dale, Jul 10 2012

A234641 Odd numbers n such that sigma(sigma(n^2)) is odd.

Original entry on oeis.org

1, 9, 38423, 104855, 247863, 313929, 345807, 376095, 469623, 623615, 787127, 943695, 985369, 1606281, 1754039, 1933815, 2034423, 2181409, 3043401, 5147241, 5545617, 5612535, 6385703, 7084143, 8868321, 10606679, 11470511, 11954409, 12276745, 12794655, 13213921, 14142695, 15512065, 15737953, 15786351, 16844135

Offset: 1

M. F. Hasler, Dec 28 2013

nonn

The sum of divisors of a square is always odd, therefore these numbers have the property that x=n^2, y=sigma(x) and z=sigma(y) are all three odd.

This is the subsequence of odd terms of A008847.

M. F. Hasler, Table of n, a(n) for n = 1..122

Cf. A000203, A000290, A008848, A163763.

A163764 a(n) = sqrt(sigma(2m^2)), where m = A097023(n), i.e., sigma(2m^2) is a square.

Original entry on oeis.org

543, 651, 5187, 5973, 7161, 8463, 57057, 93093, 66063, 81003, 80199, 98553, 130851, 160797, 216657, 259749, 347529, 561393, 565383, 726693, 882189, 1042587, 1084083, 922467, 1439361, 1242927, 1768767, 1490139, 2383227, 2857239, 2029143, 2486169, 4517877, 6175323

Offset: 1

M. F. Hasler, Aug 03 2009

nonn

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

Cf. A000203, A008847, A097023, A163763.

Select[Table[Sqrt[DivisorSigma[1, 2 n^2]], {n, 100000}], IntegerQ[#] &] (* Tanya Khovanova, Jun 18 2021 *)

a(31)-a(34) from Amiram Eldar, Aug 13 2024

A063900 Numbers k such that sum of proper divisors or aliquot parts of k^2 (excluding 1) is a square, or A048050(k^2) is a square.

Original entry on oeis.org

866, 9271, 18167, 30887, 39959, 114607, 119279, 129911, 153631, 239111, 343207, 517591, 582583, 602159, 607340, 1202282, 1397863, 1729999, 1920647, 2533183, 2547119, 2558183, 5740127, 7122959, 9379871, 10218407, 10891103, 13549399

Offset: 1

Jason Earls, Aug 29 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100 (terms 1..55 from Harry J. Smith)
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A008847, A048050.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := IntegerQ@Sqrt@chowla[n^2]; Select[Range[10^6], aQ] (* Amiram Eldar, Aug 30 2019 *)

s(n)=sigma(n)-n-1;
for(n=1,10^8, if(issquare(s(n^2)),print(n)))

s(n)=sigma(n) - n - 1
{ n=0; for (m=1, 10^9, if(issquare(s(m^2)), write("b063900.txt", n++, " ", m); if (n==55, break)) ) } \\ Harry J. Smith, Sep 02 2009

a(19)-a(28) from Harry J. Smith, Sep 02 2009

A064498 Numbers k such that the sum of unitary divisors of k^2 is a square.

Original entry on oeis.org

1, 42, 120, 156, 246, 287, 1434, 1673, 2016, 5256, 9799, 11808, 18330, 19740, 21385, 34440, 39990, 44772, 45990, 46655, 57270, 60156, 66815, 68832, 102648, 115620, 125255, 149472, 156570, 170820, 182665, 195510, 200760, 208182, 223944, 224196

Offset: 1

Jason Earls, Oct 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..100 from Harry J. Smith)

Cf. A034448 (usigma), A008847 (similar, with sigma).

sudsQ[n_]:=Module[{uds=Sort[Flatten[Outer[Times,Sequence@@({1,#}&/@ Power@@@FactorInteger[n^2])]]]},IntegerQ[Sqrt[Total[uds]]]]; Join[{1}, Select[Range[230000],sudsQ]] (* Harvey P. Dale, Dec 09 2011 *)

{usigma(n, s=1, fac, i)= fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]); ); return(s); }
for(n=1,10^6, if(issquare(usigma(n^2)),print1(n," ")))

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, if (issquare(usigma(m^2)), write("b064498.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 16 2009

A097025 Numbers n such that both sigma(n) and sigma(sigma(n)) are odd numbers.

Original entry on oeis.org

1, 81, 400, 32400, 195938, 224450, 1705636, 3648100, 13645088, 15870978, 18180450, 29184800, 138156516, 295496100, 1055340196, 1105252128, 1476326929, 2263475776, 2323432804, 2363968800, 2592846400, 2661528100, 2686005218, 2917410498, 3564550178

Offset: 1

Labos Elemer, Aug 24 2004

nonn

Only members of A028982 are candidates. - Robert G. Wilson v, Aug 27 2004

Question: iterating sigma()=A000203, how many iterates can be odd numbers?

			n = 1910^2, sigma(n) = 2821^2, sigma(sigma(n)) = 10357983.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A008847, A000203, A097024, A028982.

t = Sort[ Flatten[ Table[{n^2, 2n^2}, {n, 36650}]]]; a = {}; Do[ If[ OddQ[ DivisorSigma[1, DivisorSigma[1, t[[n]] ]]], AppendTo[a, t[[n]] ]], {n, 2*10^6}]; a (* Robert G. Wilson v, Aug 27 2004 *)

Edited and extended by Robert G. Wilson v, Aug 27 2004

A232445 Numbers n such that sigma(n) and sigma(n^2) are squares.

Original entry on oeis.org

1, 11177320, 182937066, 159839399818, 166474679436

Offset: 1

Alex Ratushnyak, Nov 24 2013

Intersection of A006532 and A008847.

sigma(a(6)) >= 10^12. - Hiroaki Yamanouchi, Sep 26 2014

			sigma(a(2)) = (64*9*5*2)^2 and sigma(a(2)^2) = (3*7*13*19*31*127)^2.
sigma(a(3)) = (64*9*5*7)^2 and sigma(a(3)^2) = (3*7*13*37*61*499)^2.
sigma(a(4)) = (256*9*5*47)^2 and sigma(a(4)^2) = (3*49*13*19*37*43*61*67)^2.
sigma(a(5)) = (16*3*7*121*17)^2 and sigma(a(5)^2) = (3*49*13*31*61*109*757)^2.

Cf. A000203 (sigma: sum of divisors of n), A006532, A008847, A232444.

isok(n) = issquare(sigma(n)) && issquare(sigma(n^2)); \\ Michel Marcus, Sep 24 2014

a(4) from Hiroaki Yamanouchi, Sep 24 2014

a(5) from Hiroaki Yamanouchi, Sep 26 2014

A063893 Numbers k such that the sum of proper divisors or aliquot parts of k^2 is a square.

Original entry on oeis.org

1, 3, 49, 35713, 102851, 949818597, 1070640001, 74145779101, 138452510557, 158236686397, 640606652893, 2814821518321, 9081196437853, 36236716613821, 97748813502577, 185178952282141, 433755354387133

Offset: 1

Jason Earls, Aug 29 2001

Numbers 640606652893, 2814821518321, 9081196437853, 36236716613821 and 97748813502577 are also terms. - Donovan Johnson, Mar 31 2012

747834604942753, 1543596825805057, 2356096433066461, 6853335405451201 and 7704992936528497 are also terms. - Martin Ehrenstein, Jul 30 2023

			49 is a term because 49^2 = 2401 and sum of aliquot divisors of 2401: 1+7+49+343 = 400 = 20^2.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.

Cf. A008847, A001065, A065764.

s(n)=sigma(n)-n;
for(n=1,10^8, if(issquare(s(n^2)), print1(n, ", ")))

a(6) from Naohiro Nomoto, Jun 06 2002
a(7) from Giovanni Resta, Jan 31 2012
a(8)-a(10) from Donovan Johnson, Mar 15 2012
a(11)-a(17) from Martin Ehrenstein, Jul 30 2023

cp's OEIS Frontend

A163763 Sqrt(sigma(A008847(n)^2)), where A008847 lists m such that sigma(m^2) is a square.

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A008848 Squares whose sum of divisors is a square.

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A097023 Numbers k such that the sum of the divisors of 2*k^2 is a square.

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A234641 Odd numbers n such that sigma(sigma(n^2)) is odd.

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A163764 a(n) = sqrt(sigma(2*m^2)), where m = A097023(n), i.e., sigma(2*m^2) is a square.

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A063900 Numbers k such that sum of proper divisors or aliquot parts of k^2 (excluding 1) is a square, or A048050(k^2) is a square.

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A064498 Numbers k such that the sum of unitary divisors of k^2 is a square.

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A097025 Numbers n such that both sigma(n) and sigma(sigma(n)) are odd numbers.

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A163764 a(n) = sqrt(sigma(2m^2)), where m = A097023(n), i.e., sigma(2m^2) is a square.