A352743 a(n) = Product_{k=1..n} (p(k+1) + p(k))/(p(k+1) - p(k)), where p(k) = prime(k).

1, 5, 20, 120, 540, 6480, 48600, 874800, 9185400, 79606800, 2388204000, 27066312000, 527793084000, 22167309528000, 498764464380000, 8312741073000000, 155171166696000000, 9310270001760000000, 198619093370880000000, 6852358721295360000000, 493369827933265920000000

Offset: 0

Author

Thomas Ordowski, Apr 01 2022

Keywords

• nonn

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

Examples

			a(4) = ((3+2)/(3-2))*((5+3)/(5-3))*((7+5)/(7-5))*((11+7)/(11-7)) = 540.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..400
• Mauro Fiorentini, Ordowski (congettura di) (in Italian).

Crossrefs

Cf. A000040, A001223, A001043, A014574, A024675, A028334, A081411, A386311.

Programs

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 *)

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

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

Formula

a(n) = Product_{k=1..n} A001043(k)/A001223(k).

a(n+1) = 5 * Product_{k=1..n} A024675(k)/A028334(k+1).

Note that A024675(k) and A028334(k+1) are relatively prime.

For n >= 2, a(n) <= (prime(n)+1)*a(n-1). - Thomas Ordowski, Jul 30 2025

Extensions

More terms from Amiram Eldar, Apr 01 2022