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

1,1

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

Links

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

Example

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.

Prog

(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++<a.length-1)
if (a[i]==a[i+1]) a.splice(i--, 1);
document.writeln(a);

Crossrefs

Cf. A000203, A001157, A185208.

Sequence in context: A055009 A054982 A108785 * A237386 A259294 A050507

Adjacent sequences: A219030 A219031 A219032 * A219034 A219035 A219036

Keyword

nonn

Author

Jon Perry, Nov 10 2012

Extensions

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

Status

approved