A375028 -id:A375028 - 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

A373798 Divide A374965 into "blocks" by saying that each prime term ends a block; sequence gives lengths of successive blocks.

Original entry on oeis.org

2, 3, 4, 5, 5, 3, 3, 11, 2, 5, 4, 4, 5, 9, 7, 2, 2, 21, 3, 2, 3, 2, 3, 2, 3, 7, 3, 3, 2, 108, 17, 5, 12, 12, 2, 4, 3, 3, 4, 16, 9, 6, 231, 3, 2, 8, 3, 29, 3, 2, 5, 10, 5, 17, 2, 3, 212, 9, 3, 4, 5, 22, 3, 5, 13, 5, 9, 4, 12, 8, 2, 57, 2, 65, 5, 3, 93, 9, 46

Offset: 1

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

nonn

The first 286 terms of the sequence are the result of dividing the first 10000 terms of A374965 into "blocks."
Comment from N. J. A. Sloane, Aug 09 2024 (Start):
Suppose p = A374965(t) is a prime in A374965, and is the s-th prime to appear there (that is, A375028(s) = p and A373799(s) = t). The next term in A374965 is by definition A374965(t+1) = prime(t+1) - 1 = r (say). Then the block starting with r has length a(s+1) = A050412(r) + 1. For example, p = 19 = A374965(5) is the second prime in A374695, so we have s = 2, t = 5, and r = prime(6) - 1 = 13 - 1 = 12. Then A050412(12) = 3, which tells us that a(3) = 3 + 1 = 4. The block is [12, 25, 51, 103].
For a larger example, the s = 285th prime in A374965 is p = 160077823 = A374965(7686), so t = 7686. The next block begins with r = prime(7687) - 1 = 78282. After 39 steps of double-and-add-1 (corresponding to A050412(78282) = 39) we reach the 286th prime in A374965, A374965(7726) = 43036534378594303. (End)

			A374965 begins
1, 3/ 4, 9, 19/ 12, 25, 51, 103/ 28, 57, 115, 231, 463/ 46, 93, 187, 375, 751/ 70, 141, 283/ 82, 165, ...,
where the primes are followed by slashes, to indicate the blocks. The lengths of the initial blocks are 2, 3, 4, 5, 5, 3, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..289 [Terms 1 to 286 from Harvey P. Dale] Obtained using Harvey P. Dale's MMA program.

Cf. A374965, A375028.

nxt[{n_, a_}] := {n + 1, If[! PrimeQ[a], 2 a + 1, Prime[n + 1] - 1]}; vvv=NestList[nxt,{1,1},9999][[;;,2]]; Total/@Partition[Length/@SplitBy[vvv,PrimeQ],2] (* Harvey P. Dale, Jul 28 2024 *)

A373804 Primes in A374965 sorted into increasing order.

Original entry on oeis.org

3, 19, 103, 283, 313, 331, 463, 733, 751, 757, 1093, 1153, 1213, 1453, 1543, 1783, 2083, 2251, 2371, 2467, 2671, 2707, 2803, 3733, 3823, 7603, 7723, 8221, 9013, 9661, 14983, 15277, 15607, 16363, 16381, 16843, 17923, 19483, 20287, 21061, 22093, 23173, 24421, 24841, 25903, 27211, 28411

Offset: 1

N. J. A. Sloane, Aug 08 2024

nonn

Since we know the first 350199 terms of A374965, and A374965(350199) = 5026186 starts a new doubling chain, we know that any subsequent prime is greater than 5026186. This implies that the terms in the b-file, which are < 5026186, are correct. Of course, if the sequence reaches a Riesel number (cf. A076337) there will be no more primes after that point.

Note that, as can be seen from the b-file in A375028, A374965 contains many primes greater than 5026186 among the first 350199 terms, including one prime with 102410 digits. But these large primes cannot be added to the present b-file until more is discovered about primes following term 350199.

N. J. A. Sloane, Table of n, a(n) for n = 1..184
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. A374965, A375028, A373799, A076337.

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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A373799 Index of n-th prime in A374965.

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A373798 Divide A374965 into "blocks" by saying that each prime term ends a block; sequence gives lengths of successive blocks.

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A373804 Primes in A374965 sorted into increasing order.

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