A263879 Length k of the longest chain of primes p_1, p_2, ..., p_k such that p_1 is the n-th prime and p_{i+1} equals 2*p_i + 1 or 2*p_i - 1 for all i < k, the +/- sign depending on i.

6, 5, 4, 2, 3, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 6, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 5, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1

Offset

1,1

Comments

If the +/- signs are all + or all -, then p_1, p_2, ..., p_k is a Cunningham chain of the first or second kind, respectively.

If p_1 > 3, then the +/- signs must be all + or all -, because if e = +1 or -1, then one of p, 2*p + e, 2*(2*p + e) - e is divisible by 3; see Löh (1989), p. 751.

Cunningham chains of the first and second kinds of length > 1 cannot begin with the same prime p > 3, because one of the numbers p, 2*p-1, 2*p+1 is divisible by 3.

References

R. K. Guy, Unsolved Problems in Number Theory, A7.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.

Wikipedia, Cunningham chain

Formula

a(n) = max(A181697(n), A181715(n)) for n > 2.

a(n) < prime(n) for n > 2; see Löh (1989), p. 751.

Example

2, 3, 5, 11, 23, 47 is the longest such chain of primes starting with 2. Their indices are 1, 2, 3, 5, 9, 15, respectively, so a(1) = 6, a(2) = 5, a(3) = 4, a(5) = 3, a(9) = 2, and a(15) = 1.

Maple

f:= proc(n) option remember; local x;
  if n mod 3 = 1 then x:= 2*n-1 else x:= 2*n+1 fi;
  if isprime(x) then 1 + procname(x) else 1 fi;
end proc:
f(2):= 6: f(3):= 5:
map(f, [seq(ithprime(i), i=1..100)]); # Robert Israel, Jul 04 2023

Mathematica

A263879 = Join[{6, 5},
  Table[p = Prime[n]; cnt = 1;
   While[PrimeQ[2*p + 1] || PrimeQ[2*p - 1],
    cnt++ && If[PrimeQ[2*p + 1], p = 2*p + 1, p = 2*p - 1 ]];
   cnt, {n, 3, 100}]]

Prog

(Python)
from sympy import prime, isprime
def A263879(n):
    if n <= 2: return 7-n
    p, c = prime(n), 1
    while isprime(p:=(p<<1)+(-1 if p%3==1 else 1)):
        c += 1
    return c # Chai Wah Wu, Jul 07 2023

Crossrefs

Cf. A005384, A005602, A181697, A181715.

Sequence in context: A084448 A256596 A173431 * A085664 A154007 A213014

Adjacent sequences: A263876 A263877 A263878 * A263880 A263881 A263882

Keyword

nonn

Author

Jonathan Sondow, Oct 28 2015

Status

approved