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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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%I #36 Aug 13 2024 14:36:31

%S 1,3,4,7,12,25,51,103,22,43,32,65,131,42,83,54,109,60,119,237,473,945,

%T 1889,90,181,100,199,108,217,435,871,1743,3487,6975,13951,27903,55807,

%U 162,323,645,1289,182,365,731,1463,2927,210,419,228,457,232,463,242,485,971,262,523,272,545,1091

%N 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.

%C 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.

%C After 4000 terms, the doubling run that began at a(2380) = 21168 wass still growing.

%C 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

%H N. J. A. Sloane, <a href="/A373805/b373805.txt">Table of n, a(n) for n = 1..4000</a>

%H Michael De Vlieger, <a href="/A373805/a373805.png">Log log scatterplot of log_10 a(n)</a>, n = 1..2^15.

%H Michael De Vlieger, <a href="/A373805/a373805.txt">Compactified table of n, a(n) = m * 2^k + b</a>, 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.

%e We start with a(1) = S = 1. Since 1 is not a prime, a(2) = 2*1 + 1 = 3.

%e 3 is a prime, so now S = -1 and a(3) = prime(3) - 1 = 5-1 = 4.

%e 4 is not a prime, so a(4) = 2*4 - 1 = 7.

%e And so on.

%p # To get the first 100 terms:

%p A:=Array(1..1200, 0);

%p t:=1;

%p A[1]:= t; S:=1;

%p for n from 2 to 100 do

%p if not isprime(t) then t:=2*t+S; else S:=-S; t:=ithprime(n)+S; fi;

%p A[n]:=t;

%p od:

%p [seq(A[n], n=1..100)];

%t 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 *)

%t 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 *)

%o (Python)

%o from sympy import sieve, isprime

%o from itertools import count, islice

%o def A373805_gen(): # generator of terms

%o an = S = 1

%o for n in count(2):

%o yield an

%o if not isprime(an): an = 2*an + S

%o else: S *= -1; an = sieve[n] + S

%o print(list(islice(A373805_gen(), 60))) # _Michael S. Branicky_, Aug 12 2024

%Y Cf. A374965, A373801, A373806, A373807.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Aug 11 2024.