# [seqfan] Re: Pairs Occurring Only Once Among # Of Divisors

Jack Brennen jfb at brennen.net
Fri Jun 26 20:46:29 CEST 2009

```T. D. Noe wrote:
>> [1, 2, 3, 4, 8, 15, 16, 24, 35, 48, 63, 64, 80, 99, 288, 528, 575, 624,
>> 728, 960, 1023, 1024, 1088, 1295, 2303, 2400, 4095, 4096, 5328, 6399,
>> 6723, 9408, 9999, 14640, 15624, 28223, 36863, 38415, 46655, 50175,
>> 50624, 57121, 59048, 59049, 65535, 65536, 71824, 82944, 83520, 117648,
>> 130320, 146688, 250000, 262143, 262144, 263168, 279840, 331775, 421200,
>> 529983, 531440, 531441, 589824, 640000, 641600, 651249, 746495, 746496,
>> 777924, 860624, 861183, 923520, 937024, 1000000]
>>
>> Some of these aren't in the sequence, they just didn't hit a match
>> before 10^8.  For instance, 1000000 is matched by 94^6, which didn't
>> take long to find.
>
> 71824 isn't in the sequence because it is matched by  868834576
> 82944 isn't in the sequence because it is matched by 1686498489
>

Another one I just found by applying a little ingenuity:

640000 isn't in the sequence.  Matched by:
718240924712219661023456897050078235203037193616

More concisely expressed as:  (16*46261^10)

(Done by searching for p such that (16*p^10+1)/169 is an integer prime.)

It seems that classifying numbers as either in the sequence or not in
the sequence is best done with a mix of human pruning and computer
search.

```