OFFSET
0,2
COMMENTS
Conjecture: a(n) is an integer for every natural n. - Thomas Ordowski
Checked up to n = 10^4. - Amiram Eldar, Mar 30 2022
Checked up to n = 10^6. - Michael S. Branicky, Apr 01 2022
Note that (a(n)-1)/(a(n)+1) is the relativistic sum of the velocities prime(k)/prime(k+1) from k = 1 to n, in units where the speed of light c = 1. - Thomas Ordowski, Apr 05 2022
a(0) = 1, a(n) is the largest k such that b(n+1) = b(n)*(k + a(n-1))/(k - a(n-1)) is prime, where b(1) = 2. By my conjecture, b(n) = prime(n). - Thomas Ordowski, Jul 30 2025
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..400
Mauro Fiorentini, Ordowski (congettura di) (in Italian).
FORMULA
EXAMPLE
a(4) = ((3+2)/(3-2))*((5+3)/(5-3))*((7+5)/(7-5))*((11+7)/(11-7)) = 540.
MAPLE
a:= proc(n) option remember; (p-> `if`(n=0, 1,
a(n-1)*(p(n+1)+p(n))/(p(n+1)-p(n))))(ithprime)
end:
seq(a(n), n=0..20); # Alois P. Heinz, Apr 01 2022
MATHEMATICA
p = Prime[Range[21]]; FoldList[Times, 1, (Rest[p] + Most[p])/(Rest[p] - Most[p])] (* Amiram Eldar, Apr 01 2022 *)
PROG
(Python)
from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
n, an, p, pp = 0, 1, 2, 3
while True:
yield an
q, r = divmod(an*(pp+p), pp-p)
assert r == 0, ("Counterexample", n, p, pp)
n, an, p, pp = n+1, q, pp, nextprime(pp)
print(list(islice(agen(), 21))) # Michael S. Branicky, Apr 01 2022
(PARI) a(n) = my(v=primes(n+1)); prod(k=1, n, (v[k+1]+v[k])/(v[k+1]-v[k])); \\ Michel Marcus, Apr 10 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Apr 01 2022
EXTENSIONS
More terms from Amiram Eldar, Apr 01 2022
STATUS
approved