cp's OEIS Frontend

A116536 Numbers that can be expressed as the ratio of the product and the sum of consecutive prime numbers starting from 2.

%I A116536 #45 Sep 08 2022 08:45:24
%S A116536 1,3,125970,1278362451795,305565807424800745258151050335,
%T A116536 2099072522743338791053378243660769678400212601239922213271230,
%U A116536 330455532167461882998265688366895823334392289157931734871641555
%N A116536 Numbers that can be expressed as the ratio of the product and the sum of consecutive prime numbers starting from 2.
%C A116536 Let prime(i) denote the i-th prime (A000040). Let F(m) = (Product_{i=1..m} prime(i)) / (Sum_{i=1..m} prime(i)). Sequence gives integer values of F(m) and A051838 gives corresponding values of m. - _N. J. A. Sloane_, Oct 01 2011
%D A116536 G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 158.
%H A116536 Amiram Eldar, <a href="/A116536/b116536.txt">Table of n, a(n) for n = 1..81</a> (terms 1..42 from Vincenzo Librandi)
%F A116536 a(n) = A002110(A051838(n)) / A007504(A051838(n)). - _Reinhard Zumkeller_, Oct 03 2011
%F A116536 a(n) = A159578(n)/A001414(A159578(n)). - _Amiram Eldar_, Nov 02 2020
%e A116536 a(1) = 1 because 2/2 = 1.
%e A116536 a(2) = 3 because (2*3*5)/(2+3+5) = 30/10 = 3.
%e A116536 a(3) = 125970 because (2*3*5*7*11*13*17*19)/(2+3+5+7+11+13+17+19) = 9699690/77 = 125790.
%p A116536 P:=proc(n) local i,j, pp,sp; pp:=1; sp:=0; for i from 1 by 1 to n do pp:=pp*ithprime(i); sp:=sp+ithprime(i); j:=pp/sp; if j=trunc(j) then print(j); fi; od; end: P(100);
%t A116536 seq = {}; sum = 0; prod = 1; p = 1; Do[p = NextPrime[p]; prod *= p; sum += p; If[Divisible[prod, sum], AppendTo[seq, prod/sum]], {50}]; seq (* _Amiram Eldar_, Nov 02 2020 *)
%o A116536 (Magma) [p/s: n in [1..40] | IsDivisibleBy(p,s) where p is &*[NthPrime(i): i in [1..n]] where s is &+[NthPrime(i): i in [1..n]]];  // _Bruno Berselli_, Sep 30 2011
%o A116536 (Haskell)
%o A116536 import Data.Maybe (catMaybes)
%o A116536 a116536 n = a116536_list !! (n-1)
%o A116536 a116536_list = catMaybes $ zipWith div' a002110_list a007504_list where
%o A116536    div' x y | m == 0    = Just x'
%o A116536             | otherwise = Nothing where (x',m) = divMod x y
%o A116536 -- _Reinhard Zumkeller_, Oct 03 2011
%Y A116536 Cf. A001414, A108552, A067111, A051838, A140763, A141092, A159578.
%Y A116536 Subsequence of A325307.
%K A116536 nonn,easy
%O A116536 1,2
%A A116536 _Paolo P. Lava_ & _Giorgio Balzarotti_, Mar 27 2006