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A283455 Numbers m such that 2^m - 1 has at most 2 distinct prime factors. 0
1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 31, 37, 41, 49, 59, 61, 67, 83, 89, 97, 101, 103, 107, 109, 127, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 521, 523, 607, 727, 809, 881, 971, 983, 997, 1061 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence differs from A283364 beginning with a(15). All a(n) > 6 are primes or squares of primes.

As in A283364 one can prove that all a(n) > 6 are odd. It is clear that a(n) is either prime or semiprime. Let us show that in the latter case it is the square of a prime. Indeed, let a(n) = p*q, p < q. Then 2^a(n)-1 is divisible by 2^p-1 < 2^q-1. Thus both of them are Mersenne primes.

Let us show that 2^(p*q)-1 differs from (2^p-1)^u*(2^q-1)^v, u,v >= 1. Indeed the equality is possible only in the case p*u + q*v = p*q. Then p|v and q|u. Let u = q*a, v = p*b. Then a + b = 1, which is impossible for u,v >= 1. Hence, 2^(p*q)-1 has a third prime divisor and p*q is not a member.

Are there terms other than 4, 9 and 49 that are squares of primes? Note that, for prime p, 2^(p^2)-1 differs from (2^p-1)^p, so if p^2 is a term, then for a Mersenne prime 2^p-1 and some t >= 1, the number (2^(p^2)-1)/(2^p-1)^t should be a prime or a power of a prime.

Numbers n such that A046800(n) < 3. - Michel Marcus, Mar 08 2017

LINKS

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

The Cunningham Project, The Main Tables, Table 2- Factorizations of 2^n-1, n odd, n<1300.

MATHEMATICA

Select[Join[Range@ 6, Range[7, 201, 2]], PrimeNu[2^# - 1] <= 2 &] (* Michael De Vlieger, Mar 08 2017 *)

PROG

(PARI) is(n)=omega(2^n-1)<3 \\ Charles R Greathouse IV, Mar 08 2017

CROSSREFS

Cf. A046800, A283364.

Union of {1}, A000043, A085724.

Sequence in context: A330500 A030151 A307360 * A283364 A131617 A214321

Adjacent sequences: A283452 A283453 A283454 * A283456 A283457 A283458

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Mar 08 2017

EXTENSIONS

More terms from Peter J. C. Moses, Mar 08 2017

a(48)-a(50) from Charles R Greathouse IV, Mar 08 2017

a(51)-a(57) from Amiram Eldar, Feb 13 2020

STATUS

approved

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Last modified December 3 01:25 EST 2022. Contains 358510 sequences. (Running on oeis4.)