A319049 Primes p such that none of p - 1, p - 2 and p - 3 are squarefree.

101, 127, 353, 727, 1277, 1423, 1451, 1667, 2153, 2351, 2647, 3187, 3251, 3511, 3701, 3719, 3727, 4421, 4951, 5051, 5393, 5527, 6427, 6653, 6959, 7517, 7867, 8527, 9127, 9551, 9803, 9851, 10243, 10253, 10487, 10831, 11273, 11351, 11777, 11827, 12007, 12251, 12277

Offset

1,1

Comments

If p is a term, so that there are primes q,r,s such that q^2|p-3, r^2|p-2 and s^2|p-1, then the sequence includes all primes == p (mod q^2*r^2*s^2). In particular, the sequence is infinite, and a(n)/(n*log(n)) is bounded above and below by constants. - Robert Israel, Sep 09 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Example

98 = 2*7^2, 99 = 3^2*11 and 100 = 2^2*5^2. So 101 is a term.

Maple

Res:= NULL: count:= 0:
p:= 1;
while count < 100 do
  p:= nextprime(p);
  if not ormap(numtheory:-issqrfree, [p-1, p-2, p-3]) then
    count:= count+1; Res:= Res, p
  fi
od:
Res; # Robert Israel, Sep 09 2018

Mathematica

Select[Prime[Range[2000]], !SquareFreeQ[# - 1] && !SquareFreeQ[# - 2] && !SquareFreeQ[# - 3]&] (* Jean-François Alcover, Sep 17 2018 *)
Select[Prime[Range[1500]], NoneTrue[#-{1, 2, 3}, SquareFreeQ]&] (* Harvey P. Dale, Apr 11 2022 *)

Prog

(PARI) isok(p) = isprime(p) && !issquarefree(p-1) && !issquarefree(p-2) && !issquarefree(p-3); \\ Michel Marcus, Sep 09 2018
(Magma) [p: p in PrimesUpTo(13000) | not IsSquarefree(p-1) and not IsSquarefree(p-2) and not IsSquarefree(p-3)]; // Vincenzo Librandi, Sep 17 2018

Crossrefs

Cf. A000040, A039787, A049231, A240473, A257545, A318959.

Sequence in context: A095635 A060916 A075793 * A052086 A154270 A056730

Adjacent sequences: A319046 A319047 A319048 * A319050 A319051 A319052

Keyword

nonn

Author

Seiichi Manyama, Sep 08 2018

Status

approved