A259307 Numbers that belong to at least one amicable multiset.

1, 6, 28, 120, 220, 284, 496, 672, 1184, 1210, 1560, 1740, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 7380, 7776, 8128, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 19260, 20448, 20640, 20664, 21168, 21384, 21924, 22200, 22428, 22752

Offset: 1

Author

Jeppe Stig Nielsen, Jun 23 2015

Keywords

• nonn

Comments

Call a finite multiset {x_1, x_2, ..., x_k} of natural numbers (the x_i need not be distinct) an amicable multiset iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k.

By definition, A255215 is a subset because a set can be regarded as a special multiset.

Also A007691 is a subset, since a k-perfect number corresponds to an amicable multiset in an obvious way. For example, since 120 is 3-perfect, the multiset {120, 120, 120} is amicable.

The first amicable multiset that belongs to neither A255215 nor A007691 is {1740, 1740, 1560}.

Links

• Jeppe Stig Nielsen, List of all amicable multisets with a sigma value below 10^7.

Crossrefs

Cf. A255215, A007691, A259302, A259303, A259304, A259305, A259306.

Programs

PARI

/* write amicable multisets */ sMax=10^7;sigmaVals=vector(sMax,x,[]);for(n=1,sMax,s=sigma(n);s<=sMax&sigmaVals[s]=concat(sigmaVals[s],[n]));(MultisetSum(numbers,desiredSum,track)=if(desiredSum<0,return);if(desiredSum==0,print(apply(x->numbers[x],track));return);for(i=if(track,track[#track],1),#numbers,MultisetSum(numbers,desiredSum-numbers[i],concat(track,[i]))));for(s=1,sMax,MultisetSum(sigmaVals[s],s,[]))