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

1,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A188764, A188717.

Sequence in context: A086218 A202637 A138658 * A067464 A081545 A274074

Adjacent sequences: A089192 A089193 A089194 * A089196 A089197 A089198

Keyword

easy,nonn

Author

Cino Hilliard, Dec 08 2003

Extensions

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

Status

approved