|
%I #12 Jul 13 2022 17:41:50
%S 2,3,467,487,787,887,1279,2063,2557,2657,2903,3323,3413,3547,3583,
%T 4273,4373,4517,4567,4801,5233,5393,5443,6047,6823,6911,7507,9133,
%U 9137,9721,9973,10103,10313,10937,12227,12763,13183,13627,14407,15073,15083,15187,15359,15787,16903,17047,17911,18013,18587
%N Primes p such that p^2-1 does not have a divisor d with d + (p^2-1)/d prime.
%C Primes p such that p^2-1 is not in A355643.
%H Robert Israel, <a href="/A355644/b355644.txt">Table of n, a(n) for n = 1..10000</a>
%e a(2) = 3 is a term because it is prime, the divisors of 3^2-1 = 8 are 1, 2, 4 and 8, and none of 1+8/1 = 9, 2+8/2 = 6, 4+8/4 = 6, 8+8/8 = 9 are prime.
%p filter:= proc(p) local n,F,t;
%p n:= p^2-1;
%p F:= select(t -> t^2 <=n, numtheory:-divisors(n));
%p not ormap(isprime, map(t -> t+n/t, F))
%p end proc:
%p select(filter, [seq(ithprime(i),i=1..3000)]);
%t q[n_] := AllTrue[Divisors[n], !PrimeQ[# + n/#] &]; Select[Prime[Range[2000]], q[#^2 - 1] &] (* _Amiram Eldar_, Jul 11 2022 *)
%o (PARI) isok(p) = isprime(p) && fordiv(p^2-1, d, if (isprime(d + (p^2-1)/d), return(0))); return(1); \\ _Michel Marcus_, Jul 11 2022
%o (Python)
%o from sympy import divisors, isprime
%o def ok(n):
%o if not isprime(n): return False
%o t = n**2 - 1
%o return not any(isprime(d+t//d) for d in divisors(t, generator=True))
%o print([k for k in range(19000) if ok(k)]) # _Michael S. Branicky_, Jul 11 2022
%Y Cf. A355643.
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Jul 11 2022
|
