A049231 -id:A049231 - cp's OEIS frontend

A049282 Primes p such that both p-2 and p+2 are squarefree.

3, 5, 13, 17, 19, 31, 37, 41, 53, 59, 67, 71, 89, 103, 107, 109, 113, 131, 139, 157, 163, 179, 181, 193, 197, 199, 211, 229, 233, 239, 251, 257, 269, 271, 283, 293, 307, 311, 337, 347, 379, 383, 397, 401, 409, 419, 431, 433, 449, 463, 467, 487, 491, 499, 503

Offset: 1

Labos Elemer

nonn

			37 is here because neither 37+2 nor 37-2 is divisible by squares.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A049231, A049233.

with(numtheory): A049282:=n->`if`(isprime(n) and issqrfree(n-2) and issqrfree(n+2), n, NULL): seq(A049282(n), n=1..10^3); # Wesley Ivan Hurt, Jun 25 2016

lst={}; Do[p=Prime[n]; If[SquareFreeQ[p-2]&&SquareFreeQ[p+2], AppendTo[lst,p]], {n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 20 2008 *)
Select[Prime[Range[100]],AllTrue[#+{2,-2},SquareFreeQ]&] (* Harvey P. Dale, Apr 18 2025 *)

lista(nn) = forprime(p=2, nn, if (issquarefree(p-2) && issquarefree(p+2), print1(p, ", "))); \\ Michel Marcus, Jun 22 2016

Intersection of A049231 and A049233.

A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

Original entry on oeis.org

29, 101, 127, 137, 149, 173, 277, 281, 317, 353, 389, 461, 509, 541, 569, 577, 641, 677, 727, 821, 857, 877, 929, 977, 1109, 1129, 1181, 1217, 1277, 1289, 1361, 1423, 1433, 1451, 1613, 1667, 1721, 1777, 1861, 1877, 1901, 1913, 1973, 2081, 2153, 2297, 2333, 2351

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			21 (= 23 - 2) is squarefree. So 23 is not a term.
27 = 3^3 and 28 = 2^2*7. So 29 is a term.

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

Cf. A000040, A039787, A049231, A240473, A257545, A319049.

[p: p in PrimesInInterval(3, 2500)| not IsSquarefree(p-2) and not IsSquarefree(p-1)]; // Vincenzo Librandi, Sep 06 2018

Select[Prime[Range[500]], !SquareFreeQ[# - 2] && !SquareFreeQ[# - 1] &] (* Vincenzo Librandi, Sep 06 2018 *)

forprime(p=2, 1e4, if(!issquarefree(p-1)&&!issquarefree(p-2), print1(p, ", "))); \\ Altug Alkan, Sep 06 2018

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

Original entry on oeis.org

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

A049229 Primes p such that p-2 is not squarefree.

Original entry on oeis.org

11, 29, 47, 83, 101, 127, 137, 149, 173, 191, 227, 263, 277, 281, 317, 353, 389, 443, 461, 479, 509, 541, 569, 577, 587, 607, 641, 659, 677, 727, 821, 827, 839, 857, 877, 911, 929, 947, 977, 983, 1019, 1031, 1091, 1109, 1129, 1163, 1181, 1217, 1277, 1289

Offset: 1

Labos Elemer

nonn

This sequence is infinite and its relative density in the sequence of the primes is equal to 1 - 2 * Product_{p prime} (1-1/(p*(p-1))) = 1 - 2 * A005596 = 0.252088... - Amiram Eldar, Feb 27 2021

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005596, A013929, A049231, A049232.

Select[Prime[Range[2,300]],!SquareFreeQ[#-2]&] (* Harvey P. Dale, Nov 14 2012 *)

isok(p) = (p>2) && isprime(p) && !issquarefree(p-2); \\ Michel Marcus, May 14 2018

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A049282 Primes p such that both p-2 and p+2 are squarefree.

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A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

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

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Maple

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A049229 Primes p such that p-2 is not squarefree.

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Author

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Mathematica

PARI