A089195 Primes p such that all prime factors of p-1 have exponent 2.

2, 5, 37, 101, 197, 677, 4357, 5477, 8837, 12101, 16901, 17957, 21317, 28901, 42437, 44101, 52901, 98597, 106277, 148997, 164837, 184901, 217157, 220901, 224677, 324901, 401957, 417317, 427717, 454277, 476101, 509797, 682277, 792101, 820837

Offset: 1

Author

Cino Hilliard, Dec 08 2003

Keywords

• easy
• nonn

Comments

This property for prime p-1 = cube only numbers does not hold since the sum of 2 cubes has factors and p-1 = q^3 => p = q^3+1 = sum of 2 cubes.

Examples

			101 is included because 100 = 2^2*5^2 only square factors. 109 is not because while 108=2^2*3^3 has a square only factor it also has a cube factor.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 2..301 from Vincenzo Librandi)

Crossrefs

Cf. A188764, A188717.

Programs

Mathematica

Prepend[Select[Table[Prime[n],{n,70000}],Length[Union[Last/@FactorInteger[#-1]]]==1&&Union[Last/@FactorInteger[#-1]]=={2}&], 2] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
seq[lim_] := Select[Select[Range[Floor[Surd[lim-1, 2]]], SquareFreeQ]^2 + 1, PrimeQ]; seq[10^6] (* Amiram Eldar, Jan 18 2025 *)

PARI

list(lim) = select(isprime, apply(x -> x^2 + 1, select(issquarefree, vector(sqrtnint(lim-1, 2), i, i)))); \\ Amiram Eldar, Jan 18 2025

Extensions

a(1) = 2 inserted by Amiram Eldar, Jan 18 2025