Rational Amicable Pair

Author

Eric W. Weisstein
February 27, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/RationalAmicablePair.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RationalAmicablePair.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Pairs of this type considered and discovered by
Yasutoshi Kohmoto

Initialization

MersennePrimeIndices

((((a+b))^n)/((a^(n-1))+(b^(n-1))))

R[n_,{a_,b_}]:=((((a+b))^n)/((a^(n-1))+(b^(n-1))))

(Subscript[R, 1])

R[1,{a,b}]

(Subscript[R, 2])

R[2,{a,b}]

(Subscript[R, 3])

R[3,{a,b}]

Type 1

a[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3, 5,5,13 ,31, 139, 277, 3877,11 ,19}]
b[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3, 5,5,13 ,31, 139, 277, 3877,239}]

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[3,{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]

Type 2

a[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3,7,11,11,17,17,19,19,23,127,307,359,3739,22433,68209,83,1931}]
b[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3,7,11,11,17,17,19,19,23,127,307,359,3739,22433,68209,162287}]

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[3,{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]

Times@@@{a[1],b[1]}

Type 3

{a,b}=Sort/@(Flatten[{Table[2,{11}],Table[3,{7}],13,17,19,19,23,41,127,227,271,541,2269,124429,Sequence@@#}]&/@{{29,569},17099})

Times@@@{a,b}

Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a,b},R[3,{Times@@a,Times@@b}]]

((((a+b))^n)/(k((a^(n-1))+(b^(n-1)))))

R[{k_,n_},{a_,b_}]:=((((a+b))^n)/(k((a^(n-1))+(b^(n-1)))))

{k,n}=={2,4}

a[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3,5,7,23,23,59,79,137,547,2477,158527,173428537,8671426849,83,1931}]
b[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,3,5,7,23,23,59,79,137,547,2477,158527,173428537,8671426849,162287}]

v4.2.1

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[{2,4},{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]

v5.0

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[{2,4},{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]

{k,n}=={6,6}

a[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,Sequence@@Table[3,{10}],5,11,13,17,23,23,23,41,43,53,53,59,89,103,107,229,409,823,1031,1801,1831,3851,4271,19751,9322471,83,1931}]
b[m_]:=Sort[{Sequence@@Table[2,{m-1}],(2^m)-1,Sequence@@Table[3,{10}],5,11,13,17,23,23,23,41,43,53,53,59,89,103,107,229,409,823,1031,1801,1831,3851,4271,19751,9322471,162287}]

v4.2

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[{6,6},{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]

v5

Do[Print[{m=MersennePrimeIndices[[i]],
Equal@@Append[ListDivisorSigma[1,{First[#],Length[#]}&/@Split[#]]&/@{a[m],b[m]},R[{6,6},{Times@@a[m],Times@@b[m]}]]}//Timing],
{i,Length[MersennePrimeIndices]}
]