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A218981 Numbers n for which sigma(n) = sigma(w) + sigma(x) + sigma(y) + sigma(z), where n = w + x + y + z, with w, x, y, z all positive.

%I A218981 #12 Dec 07 2019 21:03:59
%S A218981 5,9,14,15,20,21,22,25,26,27,28,32,33,34,35,38,39,40,42,44,45,46,49,
%T A218981 50,51,52,54,55,56,57,58,62,63,64,65,66,68,69,70,74,75,76,77,78,80,81,
%U A218981 82,85,86,87,88,91,92,93,94,95,96,98,99,100
%N A218981 Numbers n for which sigma(n) = sigma(w) + sigma(x) + sigma(y) + sigma(z), where n = w + x + y + z, with w, x, y, z all positive.
%C A218981 Conjecture: This sequence is infinite.
%e A218981 1 + 1 + 3 + 4 = 9 and sigma(1) + sigma(1) + sigma(3) + sigma(4) = 1 + 1 + 4 + 7 = 13 = sigma(9).
%o A218981 (JavaScript)
%o A218981 function divisorSum(n) {
%o A218981 c=0;
%o A218981 for (i=1;i<=n;i++) if (n%i==0) c+=i;
%o A218981 return c;
%o A218981 }
%o A218981 ds=new Array();
%o A218981 for (j=1;j<401;j++) ds[j]=divisorSum(j);
%o A218981 a=new Array();
%o A218981 ac=0;
%o A218981 for (j=1;j<100;j++)
%o A218981 for (k=1;k<=j;k++)
%o A218981 for (m=1;m<=k;m++)
%o A218981 for (n=1;n<=m;n++)
%o A218981 if (ds[j]+ds[k]+ds[m]+ds[n]==ds[j+k+m+n]) a[ac++]=j+k+m+n;
%o A218981 a.sort(function(a, b) {return a-b;});
%o A218981 i=0;
%o A218981 while(i++<a.length-1)
%o A218981 if (a[i]==a[i+1]) a.splice(i--,1);
%o A218981 document.writeln(a);
%Y A218981 Cf. A000203, A211223, A218852, A218980.
%K A218981 nonn
%O A218981 1,1
%A A218981 _Jon Perry_, Nov 08 2012