A373807 -id:A373807 - cp's OEIS frontend

A373805 If a(n-1) is not a prime, then a(n) = 2*a(n-1) + S; otherwise set S = -S and a(n) = prime(n) + S; start with a(1) = S = 1.

1, 3, 4, 7, 12, 25, 51, 103, 22, 43, 32, 65, 131, 42, 83, 54, 109, 60, 119, 237, 473, 945, 1889, 90, 181, 100, 199, 108, 217, 435, 871, 1743, 3487, 6975, 13951, 27903, 55807, 162, 323, 645, 1289, 182, 365, 731, 1463, 2927, 210, 419, 228, 457, 232, 463, 242, 485, 971, 262, 523, 272, 545, 1091

Offset: 1

N. J. A. Sloane, Aug 11 2024

nonn

The doubling and adding-or-subtracting 1 runs alternate between Riesel type (as in A374965) and Sierpinski type (as in A373801). The interest, as in both of those sequences, is whether the sequence will hit a Riesel or Sierpinski number. If that ever happens, from that point on the sequence will double and add +-1 for ever and no more primes will appear.
After 4000 terms, the doubling run that began at a(2380) = 21168 wass still growing.
This doubling run finally terminated at a(8475) = 21167 * 2^6095 + 1. See link in A373806 for decimal expansion. - Michael De Vlieger, Aug 12 2024

			We start with a(1) = S = 1. Since 1 is not a prime, a(2) = 2*1 + 1 = 3.
3 is a prime, so now S = -1 and a(3) = prime(3) - 1 = 5-1 = 4.
4 is not a prime, so a(4) = 2*4 - 1 = 7.
And so on.

N. J. A. Sloane, Table of n, a(n) for n = 1..4000
Michael De Vlieger, Log log scatterplot of log_10 a(n), n = 1..2^15.
Michael De Vlieger, Compactified table of n, a(n) = m * 2^k + b, n = 1..10^5, where k is the 2-adic valuation of a(n)-1 if a(n) is odd, or a(n) if a(n) is even, b = a(n) mod 2, and m = (a(n)-b)/2^k.

Cf. A374965, A373801, A373806, A373807.

# To get the first 100 terms:
A:=Array(1..1200, 0);
t:=1;
A[1]:= t; S:=1;
for n from 2 to 100 do
if not isprime(t) then t:=2*t+S; else S:=-S; t:=ithprime(n)+S; fi;
A[n]:=t;
od:
[seq(A[n], n=1..100)];

nn = 120; s = j = 1; {1}~Join~Reap[Do[If[PrimeQ[j], s = -s; k = Prime[n] + s, k = 2 j + s]; j = k; Sow[k], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Aug 11 2024 *)
m = 120; ToExpression /@ Import["https://oeis.org/A373805/a373805.txt", "Data"][[;; m, -1]] (* Generate up to m = 10^5 terms from compactified a-file, Michael De Vlieger, Aug 13 2024 *)

from sympy import sieve, isprime
from itertools import count, islice
def A373805_gen(): # generator of terms
    an = S = 1
    for n in count(2):
        yield an
        if not isprime(an): an = 2*an + S
        else: S *= -1; an = sieve[n] + S
print(list(islice(A373805_gen(), 60))) # Michael S. Branicky, Aug 12 2024

A373806 Primes in A373805 in order of their occurrence.

Original entry on oeis.org

3, 7, 103, 43, 131, 83, 109, 1889, 181, 199, 55807, 1289, 2927, 419, 457, 463, 971, 523, 1091, 563, 617, 159233, 761, 1549, 821, 839, 6959, 919, 937, 971, 2003, 1039, 17793023, 1279, 1297, 10513, 11087, 5849, 12143, 3181, 1685503, 3541, 58943, 3877, 15887, 2039, 2069, 8377, 2141, 2179, 2304770047

Offset: 1

N. J. A. Sloane, Aug 11 2024

nonn

The next prime after 169159 is not presently known.
Added Aug 14 2024: it is 123287*2^m + 1, where m = 2538167. See the Ballinger-Keller link. More details will be added soon. - N. J. A. Sloane, Aug 14 2024
From Michael De Vlieger, Aug 12 2024: (Start)
See link for decimal expansion of a(320) = 21167 * 2^6095 + 1, a number with 1840 decimal digits.
a(607) = A373805(11585) = 1971503; no other primes seen for n <= 2^16. (End)

N. J. A. Sloane, Table of n, a(n) for n = 1..319 [a(320) is too large to include - see _Michael De Vlieger_'s comment and links]
Ray Ballinger and Wilfrid Keller, The Sierpiński Problem: Definition and Status
Michael De Vlieger, Decimal expansion of A373805(8475) = a(320), bfile format with offset 1.
Michael De Vlieger, Table of n, a(n) for n = 321..607

Cf. A373805, A373807.

nn = 2^14; s = j = 1; Reap[Monitor[Do[If[PrimeQ[j], Sow[j]; s = -s; k = Prime[n] + s, k = 2 j + s]; j = k, {n, 2, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Aug 12 2024 *)

# uses imports and A373805_gen in A373805
def agen(): # generator of terms
    yield from (ai for i, ai in enumerate(A373805_gen(), 1) if isprime(ai))
print(list(islice(agen(), 51))) # Michael S. Branicky, Aug 12 2024

A373808 If a(n-1) is not a prime, then a(n) = 2*a(n-1) + S; otherwise set S = -S and a(n) = prime(n) + S; start with a(1) = 2, S = -1.

Original entry on oeis.org

2, 4, 9, 19, 10, 19, 18, 37, 22, 43, 32, 65, 131, 42, 83, 54, 109, 60, 119, 237, 473, 945, 1889, 90, 181, 100, 199, 108, 217, 435, 871, 1743, 3487, 6975, 13951, 27903, 55807, 162, 323, 645, 1289, 182, 365, 731, 1463, 2927, 210, 419, 228, 457, 232, 463, 242, 485, 971, 262, 523, 272, 545, 1091, 282

Offset: 1

N. J. A. Sloane, Aug 12 2024

nonn

Similar to A373805, but with different initial values. Contains a repeated term (19), but that is allowed. Merges with A373805 at a(10) = 22. The primes here are therefore essentially the same as the primes in A373805: see A373806 and A373807.

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Cf. A373805, A373806, A373807.

Module[{n = 1, s = -1}, NestList[If[n++; PrimeQ[#], Prime[n] + (s = -s), 2*# + s] &, 2, 100]] (* Paolo Xausa, Aug 13 2024 *)

cp's OEIS Frontend

A373805 If a(n-1) is not a prime, then a(n) = 2*a(n-1) + S; otherwise set S = -S and a(n) = prime(n) + S; start with a(1) = S = 1.

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A373806 Primes in A373805 in order of their occurrence.

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Python

A373808 If a(n-1) is not a prime, then a(n) = 2*a(n-1) + S; otherwise set S = -S and a(n) = prime(n) + S; start with a(1) = 2, S = -1.

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Mathematica