amicable numbers shyam sunder gupta introduction distribution amicable number fascinating observation harshad amicable pairs happy amicable pairs amicable numbers introduction distribution amicable number fascinating observation harshad amicable pairs happy amicable pairs introduction pythagoras ask friend reply friend number form small pair amicable number know friendly number know pythagoras number call amicable friendly equal sum aliquot divisor aliquot divisor mean divisor exclude number example aliquot divisor number aliquot divisor number represent amicable pair ap m n sum aliquot divisor m n s m s n respectively amicable pair s m s n s n s m see amicable number power generate symbolize mutual harmony perfect friendship love small amicable pair know antiquity significance attach possessor assure close friendship possessor number pair marriage past basis amicable number great pierre de fermat discover pair amicable number later descartes give pair amicable number result actually rediscovery number know arab mathematician th century great euler draw list amiable pair later show unfriendly paganini year old italian startle mathematical world announce number friendly second low pair completely overlook eulers list amicable pairs contain sept pair amicable number know rule find amicable number set contrast single formula perfect number simple rule give arab thabit ibn korrah th century give power n form number h n t s h t s prime h ht n s amicable example prime n h t n s amicable number similarly n amicable pair distribution amicable numbers amicable number pair compile publish te riele amicable pair subsequently amicable pairs small member find detail know amicable pairs find distribution amicable pairs give table table distribution amicable pair x amicable pairs small number x look distribution amicable pair observe number amicable pair aps small number n y number amicable pair small number shall 2y n positive integer number great example number amicable pair minimum number amicable pair correct actual number amicable pair see observation hold good number amicable pair base observation number amicable pair see total number amicable pair know fascinating observation fascinating observation amicable pair give amicable pair number square amicable pair m n sum digit m n equal example consider amicable pair sum digit sum digit example amicable pair etc number enthusiast find amicable pair amicable pairs amicable pairs list amicable pair available amicable pair m n m n divisible sum digit example consider amicable pair divisible divisible detail pair give head harshad amicable pairs minimal maximal value m n amicable pair m n m n give correspond amicable pair amicable pair minimal maximal value m n base know amicable pair present convey jan munch pedersen email date peculiar note number amicable pairs end let k unit digit m n amicable pair m n m n end note small amicable pairs note amicable pair k represent rank number amicable number 292821792417ends k small pair number pair end harshad amicable pairs harshad niven number number divisible sum digit example divisible harshad number define harshad amicable pair amicable pair m n m n harshad number example consider amicable pair divisible divisible harshad number amicable pair harshad amicable pair example etc search amicable pair good past time harshad amicable pairs amicable pairs list harshad amicable pair available happy amicable pairs iterate process sum square decimal digit number process terminate original number call happy number example define happy amicable pair amicable pair m n m n happy number example consider amicable pair happy number amicable pair happy amicable pair example etc search amicable pair good past time happy amicable pairs amicable pairs list happy amicable pairs available reference reading te riele computation available pair math comp pp supplement s9 s40 te riele borho haffmann table amicable pairs cwi report history theory numbers vol amicable number pairs triplets parardh vol home