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).

434, 2170, 4774, 5642, 7378, 8246, 9982, 10850, 12586, 16058, 17794, 18662, 20398, 23002, 23870, 25606, 26474, 28210, 29078, 30814, 31682, 34286, 36022, 36890, 38626, 41230, 42098, 43834, 44702, 47306, 49042, 49910, 52514, 54250, 55118, 56854, 59458, 60326

Offset: 1

Author

Jon Perry, Nov 10 2012

Keywords

• nonn

Comments

Conjecture: This sequence is infinite.

Conjecture: The sequence only consists of even numbers.

Conjecture: The partitions only consist of even numbers.

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

Conjecture: With the lower partition as 6*A185208(n) and the upper partition 214/3 = 71.3333... of this, then the equalities are satisfied.

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).

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

Examples

			2140 + 30 = 2170.
sigma_1(2140) + sigma_1(30) = 4536 + 72 = 4608 = sigma_1(2170).
sigma_2(2140) + sigma_2(30) = 6251700 + 1300 = 6253000 = sigma_2(2170).
Hence, 2170 is in the sequence.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A001157, A185208.

Programs

JavaScript

function divisorSum(n,x) {
c=0;
for (i=1;i<=n;i++) if (n%i==0) c+=Math.pow(i,x);
return c;
}
ds=new Array();
for (j=1;j<40001;j++) {
ds[j]=new Array();
ds[j][0]=divisorSum(j,1);
ds[j][1]=divisorSum(j,2);
}
a=new Array();
ac=0;
for (j=1;j<20000;j++)
for (k=1;k<=j;k++)
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+" ::: ";
a.sort(function(a, b) {return a-b;});
i=0;
while(i++
				

Extensions

a(6) corrected and a(13)-a(38) from Donovan Johnson, Nov 10 2012