A110928 Pairs of distinct numbers m and n, m<n, such that sigma_2(m)=sigma_2(n), where sigma_2(n) is the sum of the squares of all divisors of n.

6, 7, 24, 26, 30, 35, 40, 47, 66, 77, 78, 91, 102, 119, 114, 133, 120, 130, 120, 141, 130, 141, 136, 157, 138, 161, 150, 175, 168, 182, 174, 203, 186, 215, 186, 217, 215, 217, 222, 259, 230, 249, 246, 287, 258, 301, 264, 286, 280, 282, 280, 329, 282, 329, 318

Offset

1,1

Comments

There do not appear to be any pairs (m,n) such that sigma_k(m)=sigma_k(n) for k>2.

For sigma_3, the first pair is (184926, 194315). Other terms may be found in A131907 and A131908. See A158915.

Links

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

Formula

sigma_2(m)=sigma_2(n), m<n.

Example

sigma_2(30)=1^1+2^2+3^2+5^2+6^2+10^2+15^2+30^2=1300 and sigma_2(35)=1^2+5^2+7^2+35^2=1300.

Maple

with(numtheory); sigmap := proc(p, n) convert(map(proc(z) z^p end, divisors(n)), `+`) end; SA2:=[]: for z from 1 to 1 do for m to 1500 do M:=sigmap(2, m); for n from m+1 to 1500 do N:=sigmap(2, n); if N=M then SA2:=[op(SA2), [m, n, N]] fi od od od; SA2; select(proc(z) z[1]<=1000 end, SA2); #just to shorten it a bit

Mathematica

a[n_] := Module[{s = DivisorSigma[2, n], ans = {}}, kmax = Ceiling[Sqrt[s]]; Do[If[DivisorSigma[2, k] == s, AppendTo[ans, k]], {k, n + 1, kmax}]; ans];  s = {}; Do[v = a[n]; Do[s = Join[s, {n, v[[k]]}], {k, 1, Length[v]}], {n, 1, 400}]; s (* Amiram Eldar, Sep 08 2019 *)

Crossrefs

Cf. A001157, A002025, A002046, A063990.

Cf. A110926, A110927, A110929.

Sequence in context: A287097 A281755 A282120 * A282370 A282005 A282659

Adjacent sequences: A110925 A110926 A110927 * A110929 A110930 A110931

Keyword

nonn,tabf

Author

Walter Kehowski, Sep 23 2005

Status

approved