[seqfan] Re: R: On the divisibility of a certain linear recurrence

[Hugo Pfoertner]

Hi Peter,

I've checked through p=157 and didn't find a counterexample.  Of course
this is only an empirical observation. The factorization of a(163) is hard.

Are there any other coefficients of a linear recurrence
a(n)=c1*a(n-1)+c2*a(n-2), other than c1= 3 and c2=8, that produce a similar
behavior for n=prime?

If not, the usual objection against making sequences using arbitrary
parameters might not apply and we could create at least 3 sequences,
starting from a(2)=3
Least prime factor of a(p(n)):
3, 17, 19, 7937, 3875057, 85655881, 67, 419, 5107, 59, 28585163, 1627, 41,
563041, 283, 107, 6435362107, 1925667763, 4153, 2131,
1850491543540265361576345006181354087969129016281, 514289, 331, 37201, 193,
10041419, 619, 6430415092373729, 3923, 5744017, 5843, 1571, 36537899,
377801, 1787, 2113, 313
Largest prime factor of a(p(n)):
3, 17, 19, 7937, 3875057, 85655881, 624669523, 2207981563, 88514071931,
548466097, 3775503987139, 716442143452957321483, 1780775573699,
5811596076161, 38609629351582573090820353, 1817374561603,
143077761236233553, 1768827268814468956771,
225877944042691675469399613458636969, 13010415851368044971,
1850491543540265361576345006181354087969129016281, 3540092609483175233,
38828807550428376855712721787851, 3715881986067409049, ...
number of prime factors of a(p(n)):
1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 5, 3, 3, 6, 3, 3, 3, 5, 1, 4, 3, 6, 3,
3, 2, 3, 3, 5, 5, 5, 2, 2, 4, 7, 4,

Hugo Pfoertner

BTW: Tomorrow Don Knuth will celebrate his 3^4-th birthday ...




On Tue, Jan 8, 2019 at 10:44 AM Mason John <john.mason at lispa.it> wrote:

> a(3)=3*a(2) + 8*a(1) = 3*3 + 8*1 = 17, right? Why 9?
> john
>
> -----Messaggio originale-----
> Da: SeqFan <seqfan-bounces at list.seqfan.eu> Per conto di Peter Luschny
> Inviato: martedì 8 gennaio 2019 10:30
> A: seqfan at list.seqfan.eu
> Oggetto: [seqfan] On the divisibility of a certain linear recurrence
>
> Consider the linear recurrence
>
>     a(n) = 3*a(n-1) + 8*a(n-2) starting 1, 3, 9, ...
>
> Is it true that p prime and p not 2 or 5 implies that a(p) is squarefree?
>
> Peter
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>