# [seqfan] Re: Destinies of sums of proper divisors.

Maximilian Hasler maximilian.hasler at gmail.com
Sat Jul 4 22:12:28 CEST 2009

```I suggest a different notion of "destiny"
(or someone may find a better name),
namely, the first term in the trajectory under
s(n)=sigma(n)-n
which is not larger than its predecessor:

dest(n)={my(t);while(n<t=s(n),n=t);t}

(Associated to this, one has the number of iterations until that
happens. This better reveals some pattern (mod 6 and multiples
thereof), see below.)

Here, dest(276) = 3050  is well defined and easy to calculate.
Furthermore:
? dest(276)
%4 = 3050
? dest(%)
%5 = 2716
? dest(%)
%6 = 4972741824053642047419709382

I must admit that dest(99225) still takes some time to compute...

Maximilian

Appendix : dest(n) and time_to_dest(n) for n=1..300

vector(300,i,dest(i))
time = 0 ms.
%3 = [0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 15, 1, 10, 9, 15, 1, 11, 1, 14,
11, 14, 1, 17, 6, 16, 13, 28, 1, 45, 1, 31, 15, 20, 13, 17, 1, 22, 17,
43, 1, 45, 1, 40, 33, 26, 1, 64, 8, 43, 21, 46, 1, 45, 17, 63, 23, 32,
1, 136, 1, 34, 41, 63, 19, 45, 1, 58, 27, 40, 1, 45, 1, 40, 49, 64,
19, 45, 1, 56, 40, 44, 1, 37, 23, 46, 33, 76, 1, 45, 21, 76, 35, 50,
25, 184, 1, 73, 57, 65, 1, 393, 1, 56, 87, 56, 1, 136, 1, 106, 41,
134, 1, 393, 29, 94, 65, 62, 25, 27333, 12, 64, 45, 100, 31, 393, 1,
127, 47, 122, 1, 512, 27, 70, 105, 134, 1, 1889486, 1, 37, 51, 74, 25,
45, 35, 76, 81, 118, 1, 1889486, 1, 148, 81, 134, 37, 184, 1, 82, 57,
112, 31, 71, 1, 130, 123, 86, 1,
1889486, 14, 154, 89, 136, 1, 393, 73, 37, 63, 92, 1, 12664, 1, 154,
65, 176, 43, 393, 29, 148, 131, 170, 1, 244, 1, 100, 141, 37, 1, 393,
1, 59, 71, 104, 37, 512, 47, 106, 105, 116, 31, 12664, 1, 166, 75,
110, 49, 1156, 39, 112, 77, 220, 31, 1889486, 1, 730, 178, 116, 1,
256, 1, 202, 153, 218, 1, 1889486, 53, 184, 83, 194, 1, 27333, 1, 157,
121, 190, 97, 393, 33, 232, 87, 218, 1, 1954, 35, 130, 177, 255, 1,
393, 45, 302, 129, 134, 1, 20612, 59, 214, 93, 208, 1, 393, 1, 218,
175, 140, 97, 3050, 1, 142, 137, 730, 1, 7267, 1, 220, 195, 218, 49,
249, 18, 250, 101, 226, 1, 7267, 65, 274, 183, 152, 37, 512]

# number of iterations until this "dest" is reached

time_to_dest(n)={my(t,c=1);while(n<t=s(n),n=t;c++);c}

vector(300,i,time_to_dest(i))
= [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1,
1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 7, 1, 1, 1,
1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1,
1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 8, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 7,
1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1,
32, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2,
1, 2, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 6, 1, 3, 1, 1, 1, 15, 1, 1, 1,
1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1,
1, 2, 1, 15, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 30, 1, 5, 1, 1, 1, 2, 1,
1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6,
1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1,
7, 1, 1, 1, 4, 1, 13, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1,
1, 2]

On Sat, Jul 4, 2009 at 1:54 PM, Richard Guy<rkg at cpsc.ucalgary.ca> wrote:
> I suggest  0  as the destiny of  1
> and  1  as the destiny of  2, 3, 4, 5,
> 7, 8, 9, ..., but it's a pretty boring
> sequence.  I'd especially like to know
> a(99225).  R.
>
> On Sat, 4 Jul 2009, Tanya Khovanova wrote:
>
>> Hello SeqFans,
>>
>> I propose a new sequence: a(n) is the smallest number in the destiny of n under the operation of "sum of proper divisors".
>>
>> Two numbers have the same destiny if they have the same tail in the aliquot sequence.
>>
>> For example, if we apply the sum of proper divisors to 95, we get a sequence:
>> 95, 25, 6, 6, 6, 6, ... The destiny will be the cycle 6, 6, 6, 6, 6, ... Hence, a(95) = 6. Similarly, a(284) = a(220) = 220.
>>
>> I am not sure what this sequence should be for most numbers: 1 or 0?
>>
>> Note. This sequence might be undefined for 276.
>>
>> Tanya
>>
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>>
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>
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>
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