# [seqfan] Re: A relation between A153971 and A098828?

David Wilson dwilson at gambitcomm.com
Wed Jan 7 17:36:38 CET 2009

```I agree with Edwin Clark that A153971 is meant to be "Primes p with
2^(p-1)+3 composite" and that its values need to be corrected. When
A153971 is corrected, I suspect the present notable correlation between
the two sequences will disappear.

As far as the relationship between A098828 and A153971:

A098828 consists of the primes of form p = 2k^2+2k-1. For any number of
this form (prime or otherwise), 2^(p-1)+3 is divisible by either 7 or
13. For the singular case p = 3, we have 2^(p-1)+3 = 7 which is prime,
for all other p in A098828, 2^(p-1)+3 > 13 and is therefore composite.
Thus every element of A098828 except 3 appears in A153971.

As Edwin Clark noted, there are additional elements in A153971 that are
not in A153971, starting with 37. Because 2^(p-1)+3 grows quickly, the
density of nearby primes decreases quickly, and we should expect primes
of the form 2^(p-1)+3 to be rare for large p (as with the Mersenne
primes). In other words, we should expect most large primes to appear in
A153971, and that A099828 will be a very thin subset of A153971.

The proper relationship is therefore

A098828 is a subset of A153971 union {3}

Edwin Clark wrote:
>
> On Wed, 7 Jan 2009, Richard Mathar wrote:
>
>
>> Are these two OEIS sequences tightly related:
>>
>> http://research.att.com/~njas/sequences/?q=id:A153971|id:A098828
>>
>> 11,23,59,83,179,263,311,419,479,683,839,1103,1511,2111,2243,2663,2963,3119,4139,4703,5099,5303,5939,7079,10223,11399,12011,12323,12959,17483,19403,21011,21839,22259,24419,25763,27143,27611,28559,30011,32003,
>>
>> 3,11,23,59,83,179,263,311,419,479,683,839,1103,1511,2111,2243,2663,2963,3119,4139,4703,5099,5303,5939,7079,10223,11399,12011,12323,12959,17483,19403,21011,21839,22259,24419,25763,27143,27611,28559,30011,
>>
>>
>
>
> The sequence A153971 is incorrectly defined:
>
> A153971 Numbers p such that 2^(p-1)+3 is not prime.
>          11, 23, 59, 83, 179, 263, 311, 419, 479, 683, 839, 1103, 1511,
> 2111, 2243, 2663, 2963, 3119, 4139, 4703, 5099, 5303, 5939, 7079, 10223,
> 11399, 12011, 12323, 12959, 17483, 19403, 21011, 21839, 22259, 24419,
> 25763, 27143, 27611, 28559, 30011, 32003
>
> Note that if p = 6 then 2^(p-1)+3 = 35, but 6 is not in the
> listed numbers. Next guess: definition should be only primes p such that
> 2^(p-1)+3 is not prime, but then 37 should be in the list, yet it is not.
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>

```