cp's OEIS Frontend

A219033 Numbers n such that n = x + y, sigma_1(n) = sigma_1(x) + sigma_1(y) and sigma_2(n) = sigma_2(x) + sigma_2(y).

%I A219033 #33 Nov 16 2012 12:03:58
%S A219033 434,2170,4774,5642,7378,8246,9982,10850,12586,16058,17794,18662,
%T A219033 20398,23002,23870,25606,26474,28210,29078,30814,31682,34286,36022,
%U A219033 36890,38626,41230,42098,43834,44702,47306,49042,49910,52514,54250,55118,56854,59458,60326
%N A219033 Numbers n such that n = x + y, sigma_1(n) = sigma_1(x) + sigma_1(y) and sigma_2(n) = sigma_2(x) + sigma_2(y).
%C A219033 Conjecture: This sequence is infinite.
%C A219033 Conjecture: The sequence only consists of even numbers.
%C A219033 Conjecture: The partitions only consist of even numbers.
%C A219033 Conjecture: None satisfy sigma_3(n) = sigma_3(x) + sigma_3(y).
%C A219033 Conjecture: With the lower partition as 6*A185208(n) and the upper partition 214/3 = 71.3333... of this, then the equalities are satisfied.
%C A219033 The first 12 partitions are (428, 6), (2140, 30), (4708, 66), (5564, 78), (7276, 102), (8132, 114), (9844, 138), (10700, 150), (12412, 174), (15836, 222), (17548, 246), (18404, 258).
%C A219033 The first example of this ratio not being used is at a(67) = 103818 where (103554, 264) satisfies the equalities. Here the ratio is 1569/4 = 392.25. - _Donovan Johnson_, Nov 13 2012
%H A219033 Donovan Johnson, <a href="/A219033/b219033.txt">Table of n, a(n) for n = 1..1000</a>
%e A219033 2140 + 30 = 2170.
%e A219033 sigma_1(2140) + sigma_1(30) = 4536 + 72 = 4608 = sigma_1(2170).
%e A219033 sigma_2(2140) + sigma_2(30) = 6251700 + 1300 = 6253000 = sigma_2(2170).
%e A219033 Hence, 2170 is in the sequence.
%o A219033 (JavaScript)
%o A219033 function divisorSum(n,x) {
%o A219033 c=0;
%o A219033 for (i=1;i<=n;i++) if (n%i==0) c+=Math.pow(i,x);
%o A219033 return c;
%o A219033 }
%o A219033 ds=new Array();
%o A219033 for (j=1;j<40001;j++) {
%o A219033 ds[j]=new Array();
%o A219033 ds[j][0]=divisorSum(j,1);
%o A219033 ds[j][1]=divisorSum(j,2);
%o A219033 }
%o A219033 a=new Array();
%o A219033 ac=0;
%o A219033 for (j=1;j<20000;j++)
%o A219033 for (k=1;k<=j;k++)
%o A219033 if (ds[j][0]+ds[k][0]==ds[j+k][0] && ds[j][1]+ds[k][1]==ds[j+k][1]) a[ac++]=j+", "+k+" ::: ";
%o A219033 a.sort(function(a, b) {return a-b;});
%o A219033 i=0;
%o A219033 while(i++<a.length-1)
%o A219033 if (a[i]==a[i+1]) a.splice(i--,1);
%o A219033 document.writeln(a);
%Y A219033 Cf. A000203, A001157, A185208.
%K A219033 nonn
%O A219033 1,1
%A A219033 _Jon Perry_, Nov 10 2012
%E A219033 a(6) corrected and a(13)-a(38) from _Donovan Johnson_, Nov 10 2012