A219033 - OEIS

%I #33 Nov 16 2012 12:03:58

%S 434,2170,4774,5642,7378,8246,9982,10850,12586,16058,17794,18662,

%T 20398,23002,23870,25606,26474,28210,29078,30814,31682,34286,36022,

%U 36890,38626,41230,42098,43834,44702,47306,49042,49910,52514,54250,55118,56854,59458,60326

%N 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 Conjecture: This sequence is infinite.

%C Conjecture: The sequence only consists of even numbers.

%C Conjecture: The partitions only consist of even numbers.

%C Conjecture: None satisfy sigma_3(n) = sigma_3(x) + sigma_3(y).

%C 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 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 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 Donovan Johnson, <a href="/A219033/b219033.txt">Table of n, a(n) for n = 1..1000</a>

%e 2140 + 30 = 2170.

%e sigma_1(2140) + sigma_1(30) = 4536 + 72 = 4608 = sigma_1(2170).

%e sigma_2(2140) + sigma_2(30) = 6251700 + 1300 = 6253000 = sigma_2(2170).

%e Hence, 2170 is in the sequence.

%o (JavaScript)

%o function divisorSum(n,x) {

%o c=0;

%o for (i=1;i<=n;i++) if (n%i==0) c+=Math.pow(i,x);

%o return c;

%o }

%o ds=new Array();

%o for (j=1;j<40001;j++) {

%o ds[j]=new Array();

%o ds[j][0]=divisorSum(j,1);

%o ds[j][1]=divisorSum(j,2);

%o }

%o a=new Array();

%o ac=0;

%o for (j=1;j<20000;j++)

%o for (k=1;k<=j;k++)

%o 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 a.sort(function(a, b) {return a-b;});

%o i=0;

%o while(i++<a.length-1)

%o if (a[i]==a[i+1]) a.splice(i--,1);

%o document.writeln(a);

%Y Cf. A000203, A001157, A185208.

%K nonn

%O 1,1

%A _Jon Perry_, Nov 10 2012

%E a(6) corrected and a(13)-a(38) from _Donovan Johnson_, Nov 10 2012