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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A084448 A256596 A173431 * A085664 A154007 A213014
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Oct 28 2015
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)