# relation between A002981,A090660,A002982,A090661

Wed Mar 3 14:31:00 CET 2004

```Dear Seqfan,

Can you prove or disprove that

(1) A090660(n) - 1 = A002981(n+3)
And
(2) A090661(n) - 1 include in A002982

we have precprime(n!) < n! < nextprime(n!)
==> (n+1)*precprime(n!) < (n+1)! < (n+1)*nextprime(n!)
and precprime((n+1)!)< (n+1)!
I compare (n+1)*nextprime(n!) with nextprime((n+1)!)
and (n+1)!*precprime(n!) with precprime((n+1)!)

from (1) n!+1 is prime ==> nextprime((n+1)!) > (n+1)*nextprime(n!) = (n+1)*(n!+1) = (n+1)! +n+1
(Remarq : (n+1)! + 1 is not prime)

from (2) n! - 1 is prime ==> precprime((n+1)!) > (n+1)*precprime(n!)=(n+1)*(n! - 1) = (n+1)! -n -1
(Remarq : (n+1)! - 1 is not prime)

%S A090660 4,12,28,38,42,74,78,117,155,321,341,400,428,873
%N A090660 Numbers n such that n*nextprime((n-1)!)-nextprime(n!) < 0.

%S A002981 0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,1477,6380,26951
%N A002981 n! + 1 is prime.

%S A002982 3,4,6,7,12,14,30,32,33,38,94,166,324,379,469,546,974,1963,3507,3610,
%T A002982 6917,21480
%N A002982 n! - 1 is prime.

%S A090661 5,8,13,15,31,34,39,95,167,325,380,470,547
%N A090661 Numbers n such that prevprime(n!)-n*prevprime((n-1)!) < 0.

Thanks.

M.Bouayoun

```