A360053 - OEIS

%I #34 Feb 08 2023 13:14:55

%S 2,3,5,11,17,23,29,47,53,59,71,83,89,107,113,131,149,167,173,179,191,

%T 197,227,233,239,251,257,263,269,293,317,347,353,359,383,389,419,431,

%U 443,449,467,479,491,503,509,557,563,569,587,593,599,617,647,653,659,677,683

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

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

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

%e Prime 5 is a term because 2^4 - 1 = 15 = 3*5 and its sole prime factor < 5 is 3, whose exponent is 1.

%e Prime 11 is a term because 2^10 - 1 = 1023 = 3 * 11 * 31 and the exponent of 3 is 1.

%e 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.

%p P:= select(isprime,[2,seq(i,i=3..1000,2)]):

%p filter:= i -> andmap(q -> 2 &^(P[i]-1)-1 mod q^2 <> 0, P[1..i-1]):

%p P[select(filter, [$1..nops(P)])]; # _Robert Israel_, Jan 24 2023

%t 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 *)

%o (PARI) forprime(p=2, 400, forprime(div=3, p-1, if(Mod(2,div^2)^(p-1)==1, next(2))); print1(p,", "))

%o (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

%o (Python)

%o from itertools import count, islice

%o from sympy import prime

%o def A360053_gen(): # generator of terms

%o     for i in count(1):

%o         p = prime(i)

%o         if all((pow(2,p-1,prime(j)**2)-1 for j in range(1,i))):

%o             yield p

%o A360053_list = list(islice(A360053_gen(),20)) # _Chai Wah Wu_, Feb 08 2023

%K nonn

%O 1,1

%A _Alain Rocchelli_, Jan 23 2023