# [seqfan] Re: Who understands Granville numbers?

Reinhard Zumkeller reinhard.zumkeller at gmail.com
Thu Oct 28 17:45:44 CEST 2010

```with the following program I just calculated A118372(18) and A118372(19):

a118372_list = sPerfect 1 []

sPerfect :: Int -> [Int] -> [Int]
sPerfect n ss =
case compare (sum \$ filter ((== 0) . mod n) \$ takeWhile (<
n) ss) n of
LT ->   sPerfect (n+1) (n:ss)
EQ -> n:sPerfect (n+1) (n:ss)
GT ->   sPerfect (n+1) ss
-- eop.

The list ss represents the set S, mentioned in the article of De Koninck
and Ivi´c, and in Walter Kehowski's contributions.

*S_perfect> take 19 a118372_list
[6,24,28,96,126,224,384,496,1536,1792,6144,8128,14336,15872,24576,98304,114688,393216,507904]
*S_perfect>

A118372(18) = 393216 = 3 * 2^17
A118372(19) = 507904 = 31 * 2^14

The calculation for further terms would take more than 7 hours or so.

Best
Reinhard

2010/10/28 Vladimir Shevelev <shevelev at bgu.ac.il>:
> About A118372. It seems very plausible that every term has the form a(n)=(2^k(n)-1)2^m(n) (maybe, this is known). If so, then there are at least 3 accompanying sequences which are not in OEIS:
>
> for 2^k(n)-1: 3 3 7 3 63 7 3 31 3 7 3 127...
> for k(n):        2 2 3 2   6  3 2  5  2 3 2    8...
> for m(n):       1 3 2 5   1  5 7  4  9 8 11  6...
>
> Regards,
>
> ----- Original Message -----
> From: Alonso Del Arte <alonso.delarte at gmail.com>
> Date: Thursday, October 28, 2010 1:10
> Subject: [seqfan]  Who understands Granville numbers?
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>
>> So I come across the concept of Granville numbers
>> (A118372<http://www.research.att.com/~njas/sequences/A118372>)
>> in Jean-Marie de Koninck's book *Those Fascinating Numbers*. But
>> the concept
>> is giving me a headache. Is the definition of the set S
>> recursive (one
>> having to do a complete divisor tree to figure out membership)
>> or am I
>> completely misunderstanding it? I've had an easier time doing my
>> taxes.
>> Al
>>
>> P.S. Daniel and I have signed up for a couple of slots for
>> Sequence of the
>> Day in November. Any SeqFan who hasn't yet chosen a Sequence of
>> the Day is
>> invited to sign up to choose it one day in November or December.
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```