A373801 -id:A373801 - cp's OEIS frontend

A373802 Primes in A373801 in order of their appearance.

2, 7, 29, 137, 7937, 569, 179, 809, 227, 263, 557, 40193, 797, 464897, 303868936193, 3833, 16097, 4457, 2309, 4793, 4937, 10289, 2693, 11057, 3002369, 52673, 27617, 1823, 7433, 1907, 497153, 4133, 269057, 2438716790407169, 2879, 2903, 93377, 2999

Offset: 1

N. J. A. Sloane, Aug 05 2024

nonn

a(239) has 3137 decimal digits and is too long for inclusion in the b-file. - Alois P. Heinz, Aug 05 2024

Alois P. Heinz, Table of n, a(n) for n = 1..238 (first 91 terms from N. J. A. Sloane)

Cf. A373801, A373803.

b:= proc(n) option remember; (m->
     `if`(isprime(m), ithprime(n)+1, 2*m-1))(b(n-1))
    end: b(1):=2:
g:= proc(n) option remember; local k; for k from 1+g(n-1)
      while not isprime(b(k)) do od; k
    end: g(0):=0:
a:= n-> b(g(n)):
seq(a(n), n=1..38);  # Alois P. Heinz, Aug 05 2024

Reap[Module[{n = 1}, Nest[If[n++; PrimeQ[#], Sow[#];Prime[n] + 1, 2*# - 1] &, 2, 500]]][[2, 1]] (* Paolo Xausa, Aug 07 2024 *)

from itertools import count
from sympy import isprime, nextprime
def A373802_gen(): # generator of terms
    a, p = 2, 3
    for i in count(1):
        if isprime(a):
            yield a
            a = p+1
        else:
            a = (a<<1)-1
        p = nextprime(p)
A373802_list = list(islice(A373802_gen(),20)) # Chai Wah Wu, Aug 05 2024

A373803 Indices of primes in A373801.

Original entry on oeis.org

1, 3, 6, 10, 19, 23, 25, 29, 31, 33, 36, 45, 48, 60, 91, 95, 101, 105, 108, 112, 116, 121, 124, 129, 142, 149, 155, 157, 161, 163, 173, 176, 185, 227, 229, 231, 238, 240, 386, 391, 393, 398, 411, 415, 435, 439, 466, 537, 543, 565, 570, 575, 577, 588, 592, 687, 694, 696, 700, 733, 738, 759

Offset: 1

N. J. A. Sloane, Aug 05 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..442 (first 91 terms from N. J. A. Sloane)

Cf. A373801, A373802.

b:= proc(n) option remember; (m->
     `if`(isprime(m), ithprime(n)+1, 2*m-1))(b(n-1))
    end: b(1):=2:
a:= proc(n) option remember; local k; for k from 1+a(n-1)
      while not isprime(b(k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..62);  # Alois P. Heinz, Aug 05 2024

Reap[Module[{n = 1}, Nest[If[n++; PrimeQ[#], Sow[n-1]; Prime[n] + 1, 2*# - 1] &, 2, 1000]]][[2,1]] (* Paolo Xausa, Aug 07 2024 *)

from itertools import count
from sympy import isprime, nextprime
def A373803_gen(): # generator of terms
    a, p = 2, 3
    for i in count(1):
        if isprime(a):
            yield i
            a = p+1
        else:
            a = (a<<1)-1
        p = nextprime(p)
A373803_list = list(islice(A373803_gen(),20)) # Chai Wah Wu, Aug 05 2024

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.

Original entry on oeis.org

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

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A373802 Primes in A373801 in order of their appearance.

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A373803 Indices of primes in A373801.

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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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