This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
105 is in the sequence because 105 = 3*5*7 and 7 - (3 + 5) = 7 - 8 = -1; 231 is in the sequence because 231 = 3 * 7 * 11 and 11 - (3 + 7) = 11 - 10 = 1.
filter:= proc(n) local P,pmax;
P:= numtheory[factorset](n);
abs(convert(P,`+`)-2*max(P))=1
end proc;
select(filter, [$1..10000]); # Robert Israel, Jun 23 2014
fpdQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Max[f]-Total[Most[f]]==1];gpdQ[n_]:=Module[{g=Transpose[FactorInteger[n]][[1]]},Max[g]-Total[Most[g]]==-1];Union[Select[Range[2,3000],fpdQ ],Select[Range[2,3000],gpdQ ]]
dbgQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},Abs[fi[[-1]]-Total[ Most[ fi]]]==1]; Select[Range[2,1500],dbgQ] (* Harvey P. Dale, Jan 01 2020 *)
Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2. Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 which sum to 60. Exponential divisors are 6, 24 and their sum is 30. Finally 60 / 30 = 2.
with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok; for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0; for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then ok:=1; for j from 1 to nops(d) do if not type(a[j][2]/d[j][2],integer) then ok:=0; break; fi; od; if ok=1 then c:=c+b[k]; fi; fi; od; if sigma(n)=2*c then print(n); fi; od; end: P(10^9);
Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* Michael De Vlieger, Jun 16 2016 *)
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