A373798 -id:A373798 - cp's OEIS frontend

A374965 a(n) = 2*a(n-1) + 1 for a(n-1) not prime, otherwise a(n) = prime(n) - 1; with a(1)=1.

1, 3, 4, 9, 19, 12, 25, 51, 103, 28, 57, 115, 231, 463, 46, 93, 187, 375, 751, 70, 141, 283, 82, 165, 331, 100, 201, 403, 807, 1615, 3231, 6463, 12927, 25855, 51711, 103423, 156, 313, 166, 333, 667, 1335, 2671, 192, 385, 771, 1543, 222, 445, 891, 1783, 238, 477

Offset: 1

Bill McEachen, Jul 25 2024

Sequence is clearly infinite and not monotonic. Primes are sparse.
When is the next prime after n=10016 ? [Answer from N. J. A. Sloane, Aug 01 2024: The point of Bill's question is that a(10016) is the prime 838951, which is in fact the 289th prime in this sequence, as can be seen from A375028 and A373799. Thanks to the work of Lucas A. Brown (see A050412), we now know that the answer to Bill's question is that the 290th prime is the 102410-digit prime 104917*2^340181 - 1 = 5079...8783, which is a(350198). It was a very good question!]
It appears that the trajectories for different initial conditions a(1) converge to a few attractors. For all prime values and most nonprime values of a(1), the trajectories converge to the same attractor with prime 838951 at n=10016. For a(1) = 147, 295, 591, 1183, ... the trajectories converge to prime 85796863 at n=4390. For a(1) = 658, the trajectory reaches a prime with 240983 digits after 800516 steps. For a(1) = 509202, the trajectory never reaches a prime (see A050412, A052333). - Chai Wah Wu, Jul 29 2024

			a(1) = 1 is not a prime, so a(2) = 2*1+1 = 3. a(2) is a prime, so a(3) = prime(3)-1 = 4. a(4) = 2*4+1 = 9.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

The primes are listed in A375028 (see also A373798 and A373804).

Cf. A050412 and A052333.

a[n_] := a[n] = If[!PrimeQ[a[n-1]], 2*a[n-1] + 1, Prime[n]-1]; a[1] = 1; Array[a, 60] (* Amiram Eldar, Jul 25 2024 *)
nxt[{n_,a_}]:={n+1,If[!PrimeQ[a],2a+1,Prime[n+1]-1]}; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Jul 28 2024 *)

from itertools import islice
from sympy import isprime, nextprime
def A374965_gen(): # generator of terms
    a, p = 1, 3
    while True:
        yield a
        a, p = p-1 if isprime(a) else (a<<1)+1, nextprime(p)
A374965_list = list(islice(A374965_gen(),30)) # Chai Wah Wu, Jul 29 2024

A373799 Index of n-th prime in A374965.

Original entry on oeis.org

2, 5, 9, 14, 19, 22, 25, 36, 38, 43, 47, 51, 56, 65, 72, 74, 76, 97, 100, 102, 105, 107, 110, 112, 115, 122, 125, 128, 130, 238, 255, 260, 272, 284, 286, 290, 293, 296, 300, 316, 325, 331, 562, 565, 567, 575, 578, 607, 610, 612, 617, 627, 632, 649, 651, 654, 866, 875, 878

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Jul 28 2024

nonn

			The fifth prime in order of appearance in A374965 is A375028(5) = 751 = A374965(19), so a(5) = 19.

N. J. A. Sloane, Table of n, a(n) for n = 1..290 (Terms 1 through 289 were obtained using _Harvey P. Dale_'s MMA program for A373798. For a(290), see A375028.)
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A373798 (first differences), A374965, A375028.

from itertools import count, islice
from sympy import isprime, nextprime
def A373799_gen(): # generator of terms
    a, p = 1, 3
    for i in count(1):
        if isprime(a):
            yield i
            a = p-1
        else:
            a = (a<<1)+1
        p = nextprime(p)
A373799_list = list(islice(A373799_gen(),20)) # Chai Wah Wu, Jul 29 2024

cp's OEIS Frontend

A374965 a(n) = 2*a(n-1) + 1 for a(n-1) not prime, otherwise a(n) = prime(n) - 1; with a(1)=1.

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