A318959 -id:A318959 - cp's OEIS frontend

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

Seiichi Manyama, Sep 08 2018

nonn

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

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

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

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

[p: p in PrimesUpTo(13000) | not IsSquarefree(p-1) and not IsSquarefree(p-2) and not IsSquarefree(p-3)]; // Vincenzo Librandi, Sep 17 2018

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

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 *)

isok(p) = isprime(p) && !issquarefree(p-1) && !issquarefree(p-2) && !issquarefree(p-3); \\ Michel Marcus, Sep 09 2018

A319050 Primes p such that neither p + 1 nor p + 2 is squarefree.

Original entry on oeis.org

7, 23, 43, 47, 79, 97, 151, 167, 223, 241, 331, 349, 359, 367, 439, 523, 547, 619, 691, 727, 773, 823, 839, 907, 1051, 1087, 1123, 1223, 1231, 1249, 1303, 1367, 1423, 1447, 1483, 1523, 1571, 1627, 1663, 1699, 1723, 1811, 1823, 1847, 1861, 1879, 1951, 1987, 2131, 2203, 2207

Offset: 1

Seiichi Manyama, Sep 08 2018

nonn

			8 = 2^3 and 9 = 3^2. So 7 is a term.
24 = 2^3*3 and 25 = 5^2. So 23 is a term.

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

Cf. A000040, A049098, A257108, A318959, A319051.

Select[Prime[Range[400]],NoneTrue[#+{1,2},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 05 2019 *)

isok(p) = isprime(p) && !issquarefree(p+1) && !issquarefree(p+2); \\ Michel Marcus, Sep 09 2018

cp's OEIS Frontend

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

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