A360053 Primes p such that each prime < p in the prime factorization of 2^(p-1) - 1 has exponent 1.

2, 3, 5, 11, 17, 23, 29, 47, 53, 59, 71, 83, 89, 107, 113, 131, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 419, 431, 443, 449, 467, 479, 491, 503, 509, 557, 563, 569, 587, 593, 599, 617, 647, 653, 659, 677, 683

Offset

1,1

Comments

For all n > 2, a(n) is congruent to 5 mod 6.

Conjecture: 2^(a(n)-1) - 1 is always squarefree.

Links

Table of n, a(n) for n=1..57.

Example

Prime 5 is a term because 2^4 - 1 = 15 = 3*5 and its sole prime factor < 5 is 3, whose exponent is 1.
Prime 11 is a term because 2^10 - 1 = 1023 = 3 * 11 * 31 and the exponent of 3 is 1.
Prime 17 is a term because 2^16 - 1 = 65535 = 3 * 5 * 17 * 257 and its two prime factors 3 and 5 which are < 17 both have exponent 1.

Maple

P:= select(isprime, [2, seq(i, i=3..1000, 2)]):
filter:= i -> andmap(q -> 2 &^(P[i]-1)-1 mod q^2 <> 0, P[1..i-1]):
P[select(filter, [$1..nops(P)])]; # Robert Israel, Jan 24 2023

Mathematica

primes[p_] := Select[Range[p - 1], PrimeQ[#] && PowerMod[2, p - 1, #] == 1 &]; q[p_] := AllTrue[primes[p], PowerMod[2, p - 1, #^2] != 1 &]; Select[Prime[Range[124]], q] (* Amiram Eldar, Jan 23 2023 *)

Prog

(PARI) forprime(p=2, 400, forprime(div=3, p-1, if(Mod(2, div^2)^(p-1)==1, next(2))); print1(p, ", "))
(PARI) isok(p) =  if (isprime(p), my(f=factor(2^(p-1)-1, p)[, 2]); (#f==0) || (vecmax(f) == 1)); \\ Michel Marcus, Feb 08 2023
(Python)
from itertools import count, islice
from sympy import prime
def A360053_gen(): # generator of terms
    for i in count(1):
        p = prime(i)
        if all((pow(2, p-1, prime(j)**2)-1 for j in range(1, i))):
            yield p
A360053_list = list(islice(A360053_gen(), 20)) # Chai Wah Wu, Feb 08 2023

Crossrefs

Sequence in context: A040078 A045309 A103664 * A129942 A113239 A049553

Adjacent sequences: A360050 A360051 A360052 * A360054 A360055 A360056

Keyword

nonn

Author

Alain Rocchelli, Jan 23 2023

Status

approved