A153208 Primes of the form 2*p-1 where p is prime and p-1 is not squarefree.

37, 73, 193, 313, 397, 457, 541, 613, 673, 757, 1153, 1201, 1321, 1453, 1621, 1657, 1753, 1873, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2857, 2917, 3061, 3217, 3313, 4057, 4177, 4273, 4357, 4441, 4561, 4933, 5077, 5101, 5113, 5233, 5437, 5581, 5701

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

Keywords

• nonn

Comments

Subsequence of A005383.

Examples

			For p = 2 (the only case with p-1 odd), 2*p-1 = 3 is prime but p-1 = 1 is squarefree, so 3 is not in the sequence. For p = 19, 2*p-1 = 37 is prime and p-1 = 18 is not squarefree, so 37 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A013929 (nonsquarefree numbers), A005383 (numbers n such that both n and (n+1)/2 are primes), A153207, A153209, A153210.

Programs

Magma

[ q: p in PrimesUpTo(2900) | not IsSquarefree(p-1) and IsPrime(q) where q is 2*p-1 ];

Maple

R:= NULL; count:= 0: p:= 3:
while count < 100 do
  p:= nextprime(p);
  if isprime(2*p-1) and not numtheory:-issqrfree(p-1) then
     R:= R, 2*p-1; count:= count+1;
  fi
od:
R; # Robert Israel, Nov 22 2023

Mathematica

lst={}; Do[p = Prime[n]; If[ !SquareFreeQ[Floor[p/2]] && PrimeQ[Ceiling[p/2]], AppendTo[lst, p]], {n, 7!}]; lst
Select[2#-1&/@Select[Prime[Range[1000]],!SquareFreeQ[#-1]&],PrimeQ] (* Harvey P. Dale, Aug 11 2024 *)

Extensions

Edited by Klaus Brockhaus, Dec 24 2008

Mathematica updated by Jean-François Alcover, Jul 04 2013