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

1, 3, 125970, 1278362451795, 305565807424800745258151050335, 2099072522743338791053378243660769678400212601239922213271230, 330455532167461882998265688366895823334392289157931734871641555

Offset: 1

Author

Paolo P. Lava & Giorgio Balzarotti, Mar 27 2006

Keywords

• nonn
• easy

Comments

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

Examples

			a(1) = 1 because 2/2 = 1.
a(2) = 3 because (2*3*5)/(2+3+5) = 30/10 = 3.
a(3) = 125970 because (2*3*5*7*11*13*17*19)/(2+3+5+7+11+13+17+19) = 9699690/77 = 125790.
		

References

• G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 158.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..81 (terms 1..42 from Vincenzo Librandi)

Crossrefs

Cf. A001414, A108552, A067111, A051838, A140763, A141092, A159578.

Subsequence of A325307.

Programs

Haskell

import Data.Maybe (catMaybes)
a116536 n = a116536_list !! (n-1)
a116536_list = catMaybes $ zipWith div' a002110_list a007504_list where
   div' x y | m == 0    = Just x'
            | otherwise = Nothing where (x',m) = divMod x y
-- Reinhard Zumkeller, Oct 03 2011

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

Maple

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);

Mathematica

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 *)

Formula

a(n) = A002110(A051838(n)) / A007504(A051838(n)). - Reinhard Zumkeller, Oct 03 2011

a(n) = A159578(n)/A001414(A159578(n)). - Amiram Eldar, Nov 02 2020