A089306 Smallest prime of the form n + (n+1)+ (n+2)+...+(n+k), or 0 if no such prime exists.

3, 2, 3, 0, 5, 13, 7, 17, 19, 0, 11, 0, 13, 29, 31, 0, 17, 37, 19, 41, 43, 0, 23, 0, 0, 53, 0, 0, 29, 61, 31, 0, 67, 0, 71, 73, 37, 0, 79, 0, 41, 0, 43, 89, 0, 0, 47, 97, 0, 101, 103, 0, 53, 109, 0, 113, 0, 0, 59, 0, 61, 0, 127, 0, 131, 0, 67, 137, 139, 0, 71, 0, 73, 149, 151, 0, 0

Offset: 1

Author

Amarnath Murthy, Nov 01 2003

Keywords

• nonn

Comments

If n is prime a(n) = n, If n is not a prime but 2n+1 is a prime then a(n) = 2n+1 else a(n) = 0, as the difference of two triangular numbers is composite if the indices differ by more than 2. r(r+1)/2 - s(s+1)/2 is composite if r-s >2.

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000

Programs

Maple

A089306 := proc(n)
    local k;
    if not isprime(n) and not isprime(2*n+1) then
        return 0 ;
    end if;
    for k from 0 do
        p := (k+1)*(k+2*n)/2 ;
        if isprime(p) then
            return p;
        end if;
    end do:
end proc; # R. J. Mathar, Jun 06 2013

Mathematica

a[n_] := Module[{k}, If[!PrimeQ[n] && !PrimeQ[2n+1], Return[0]]; For[k = 0, True, k++, p = (k+1)(k+2n)/2; If[PrimeQ[p], Return[p]]]];
Array[a, 100] (* Jean-François Alcover, Mar 25 2020, after R. J. Mathar *)

Extensions

More terms from David Wasserman, Sep 09 2005