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

%I A352743 #62 Aug 05 2025 17:57:31
%S A352743 1,5,20,120,540,6480,48600,874800,9185400,79606800,2388204000,
%T A352743 27066312000,527793084000,22167309528000,498764464380000,
%U A352743 8312741073000000,155171166696000000,9310270001760000000,198619093370880000000,6852358721295360000000,493369827933265920000000
%N A352743 a(n) = Product_{k=1..n} (p(k+1) + p(k))/(p(k+1) - p(k)), where p(k) = prime(k).
%C A352743 Conjecture: a(n) is an integer for every natural n. - _Thomas Ordowski_
%C A352743 Checked up to n = 10^4. - _Amiram Eldar_, Mar 30 2022
%C A352743 Checked up to n = 10^6. - _Michael S. Branicky_, Apr 01 2022
%C A352743 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
%C A352743 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
%H A352743 Amiram Eldar, <a href="/A352743/b352743.txt">Table of n, a(n) for n = 0..400</a>
%H A352743 Mauro Fiorentini, <a href="http://www.bitman.name/math/article/2050">Ordowski (congettura di)</a> (in Italian).
%F A352743 a(n) = Product_{k=1..n} A001043(k)/A001223(k).
%F A352743 a(n+1) = 5 * Product_{k=1..n} A024675(k)/A028334(k+1).
%F A352743 Note that A024675(k) and A028334(k+1) are relatively prime.
%F A352743 For n >= 2, a(n) <= (prime(n)+1)*a(n-1). - _Thomas Ordowski_, Jul 30 2025
%e A352743 a(4) = ((3+2)/(3-2))*((5+3)/(5-3))*((7+5)/(7-5))*((11+7)/(11-7)) = 540.
%p A352743 a:= proc(n) option remember; (p-> `if`(n=0, 1,
%p A352743       a(n-1)*(p(n+1)+p(n))/(p(n+1)-p(n))))(ithprime)
%p A352743     end:
%p A352743 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 01 2022
%t A352743 p = Prime[Range[21]]; FoldList[Times, 1, (Rest[p] + Most[p])/(Rest[p] - Most[p])] (* _Amiram Eldar_, Apr 01 2022 *)
%o A352743 (Python)
%o A352743 from sympy import nextprime
%o A352743 from itertools import islice
%o A352743 def agen(): # generator of terms
%o A352743     n, an, p, pp = 0, 1, 2, 3
%o A352743     while True:
%o A352743         yield an
%o A352743         q, r = divmod(an*(pp+p), pp-p)
%o A352743         assert r == 0, ("Counterexample", n, p, pp)
%o A352743         n, an, p, pp = n+1, q, pp, nextprime(pp)
%o A352743 print(list(islice(agen(), 21))) # _Michael S. Branicky_, Apr 01 2022
%o A352743 (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
%Y A352743 Cf. A000040, A001223, A001043, A014574, A024675, A028334, A081411, A386311.
%K A352743 nonn
%O A352743 0,2
%A A352743 _Thomas Ordowski_, Apr 01 2022
%E A352743 More terms from _Amiram Eldar_, Apr 01 2022