# [seqfan] Re: Question from Harvey Dale about A233552

Hugo Pfoertner yae9911 at gmail.com
Tue May 28 15:25:11 CEST 2019

```Could be something like
2, 2, 1, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 2, 1, 17, 3, 1, 16, 2, 0, 2, 2, 12,
1, 3, 4, 8, 0, 1, 1, 2, 1, 2, 2, 1, 4, 33, 4, 1, 5, 42, 1, 2, 1, 7, 9, 3,
0, 5, 1, 1, 2, 6, 1, 9, 1, 1, 2, 3,
-13,
3, 4, 2, 1, 7, 3, 6, 2, 23, 1, 1, 2, 1, 1, 2, 1, 8, 3, 1, 58, 5, 648, 1, 9,
16, 2, 0, 40, 11, 2, 4, 4, 5, 1, 1, 2, 4, 4, 3, 1, 214, 2, 0, 5, 5, 1, 2,
51, 1, 2, 5, 12, 4, 3, 1, 1, 6, 1, 7, 2, 1, 88, 6, 1, 27, 6, 1, 3, 1, 2, 2,
12, 5, 2, 1, 2, 9, 0, 1, 2, 28, 1, 17, 24, 13, 6, 1, 1, 24, 1,
-13
with 0 at the positions of the squares and negative numbers at the
positions of the current terms.
e.g. a(0)=2 with j=6*n+1 j=1 leads to k=1: (1*4^1-1)/3 = 1 not prime, k=2:
(2*4^2-1)/3 = 5 prime -> a(0)=2
a(1)=2: j=7, (7*4^1-1)/3=9 not prime, (7*4^2-1)/3=37 prime
a(2)=1, j=13, (13*4^1-1)/3=17 prime
a(4)=0 j=25 square
a(60)=-13 j=361, covering set {3,5,7,13}
to be submitted?
...

On Tue, May 28, 2019 at 2:18 PM M. F. Hasler <oeis at hasler.fr> wrote:

> Maybe it could be interesting to have a "companion sequence" of "proofs",
> like e.g.,
> b(n) = the least exponent k which leads to a prime (n*4^k-1)/3,
> or -p where p is the least possible largest prime of a covering set proving
> that no such k exists for n,
> or 0 if none of the above exist (? can this happen ?).
>
> - Maximilian
>
> On Tue, May 28, 2019 at 6:43 AM Hugo Pfoertner <yae9911 at gmail.com> wrote:
>
> > Just for clarification: 3145 is eliminated by (3145*4^39870-1)/3 being a
> > (pseudo)prime. Therefore in a sequence for replacement of A233552 the
> first
> > term with unknown status is at the moment 3991 (no primes through
> k=27000).
> > 6019 and 8869 also lead to primes and may be dropped from the list of
> > numbers with unknown status.
> >
> > On Tue, May 28, 2019 at 2:36 AM David Radcliffe <dradcliffe at gmail.com>
> > wrote:
> >
> > > All of the terms listed in A233552 are correct, but many are missing.
> > Here
> > > are the correct initial terms for A233552:
> > > 25, 49, 121, 169, 289, 361, 529, 625, 841, 919, 961, 1225, 1369, 1681,
> > > 1849, 2209, 2401, 2419, 2629, 2809, 3025
> > >
> > > All numbers of the form (6k + 1)^2 or (6k - 1)^2 with k >= 1 are terms.
> > >
> > > The following numbers are terms because they have covering sets of
> prime
> > > divisors:
> > > 919, 2419, 2629, 3301, 5209, 5539, 5581, 6421, 7771, 8551, 9109, 9871,
> > > 10039, 10819, 11491, 13399, 13729, 13771, 14611, 15661, 15961, 16741,
> > > 17299, 18061, 18229, 18799, 19009, 19681, 21589, 21919, 21961, 24151,
> > > 24931, 25489
> > >
> > > The covering set for 15661 is {3,5,7,17,241}. For the rest,
> {3,5,7,11,13}
> > > is a covering set, not necessarily minimal.
> > >
> > > The following numbers have unknown status. For each n in this list, I
> > have
> > > not found k < 10000 such that (n*4^k-1)/3 is a pseudoprime, nor have I
> > > found a covering set.
> > >
> > > 3145, 3991, 5461, 6019, 7309, 8869, 12091, 14701, 18439, 19651, 20569,
> > > 21289, 21781, 22171, 22285, 22519, 22789, 22891
> > >
> >
>
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>

```