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%I #18 Feb 21 2024 01:45:33
%S 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,
%T 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,
%U 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
%N Smallest prime of the form n + (n+1)+ (n+2)+...+(n+k), or 0 if no such prime exists.
%C 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.
%H R. J. Mathar, <a href="/A089306/b089306.txt">Table of n, a(n) for n = 1..1000</a>
%p A089306 := proc(n)
%p local k;
%p if not isprime(n) and not isprime(2*n+1) then
%p return 0 ;
%p end if;
%p for k from 0 do
%p p := (k+1)*(k+2*n)/2 ;
%p if isprime(p) then
%p return p;
%p end if;
%p end do:
%p end proc; # _R. J. Mathar_, Jun 06 2013
%t 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]]]];
%t Array[a, 100] (* _Jean-François Alcover_, Mar 25 2020, after _R. J. Mathar_ *)
%K nonn
%O 1,1
%A _Amarnath Murthy_, Nov 01 2003
%E More terms from _David Wasserman_, Sep 09 2005
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