A new theorem about aliquot sequences

[science_man_88]

okay it's not as good as I thought but I can explain it using sets:

basically when trying to find (sigma(x*y)-x*y) using (sigma(x)-x) it depends on sigma(y) and the intersection of 2*divsors(x) and divisors(x):

6=[1,2,3,6] for y = 2 adds on to [2,4,6,12] which have intersection [2,6] 3*(sigma(6)-6) +6 - (2+6) = 24-8 = 16 since odd numbers don't have even divisors this problem for y=2 only shows up in even number x , other y may create problems in the odd numbered x.
12=[1,2,3,4,6,12]

sigma2(n,d)=sum(x=1,#divisors(n),sigma(d)*divisors(n)[x])-sum(y=1,#intersection(divisors(n),divisors(n)*d),intersection(divisors(n),divisors(n)*d)[y])-d*n

by the looks of it.

intersection(x,y,pretty)=C=[];for(i=1,#x,for(j=1,#y,if(x[i]==y[j],C=concat(C,x[i]))));if(pretty!=0,printVectorAsSet(vecsort(C,,8)),vecsort(C,,8))

the intersection code was something I had a lot of help on thank you to everyone involved. just realized the possible overwrite of variables situation. and I've found ones that may make errors so back to checking them.duh over simplified it other ones can intersect with y=6 we have more than x and 6*x to consider.

[science_man_88]

Originally Posted by science_man_88

okay it's not as good as I thought but I can explain it using sets:

basically when trying to find (sigma(x*y)-x*y) using (sigma(x)-x) it depends on sigma(y) and the intersection of 2*divsors(x) and divisors(x):

6=[1,2,3,6] for y = 2 adds on to [2,4,6,12] which have intersection [2,6] 3*(sigma(6)-6) +6 - (2+6) = 24-8 = 16 since odd numbers don't have even divisors this problem for y=2 only shows up in even number x , other y may create problems in the odd numbered x.
12=[1,2,3,4,6,12]

sigma2(n,d)=sum(x=1,#divisors(n),sigma(d)*divisors(n)[x])-sum(y=1,#intersection(divisors(n),divisors(n)*d),intersection(divisors(n),divisors(n)*d)[y])-d*n

by the looks of it.

intersection(x,y,pretty)=C=[];for(i=1,#x,for(j=1,#y,if(x[i]==y[j],C=concat(C,x[i]))));if(pretty!=0,printVectorAsSet(vecsort(C,,8)),vecsort(C,,8))

the intersection code was something I had a lot of help on thank you to everyone involved. just realized the possible overwrite of variables situation. and I've found ones that may make errors so back to checking them.duh over simplified it other ones can intersect with y=6 we have more than x and 6*x to consider.

okay I get where I went wrong I didn't think it would come down to factoring it but it does.