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

Author

N. J. A. Sloane, Aug 11 2024

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A374965, A373801, A373806, A373807.

Programs

Maple

# 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)];

Mathematica

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 *)

Python

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