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A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.

%I A239309 #5 Mar 18 2014 05:42:05
%S A239309 1,0,2,5,3,37,21,29,67,71,23,11,15,7,58,12,41,8,66,25,35,370,35,17,75,
%T A239309 159,198,30,37,153,232,333,170,507,108,279,41,61,486,9,194,211,29,73,
%U A239309 173,575,152,214,10,147,126,672,388,77,358,1048,528,291,322,1491
%N A239309 a(n) is the smallest k such that prime(n) divides Sum_{i=1..k} A086169(i), or 0 if no such k exists, where A086169(i) is the sum of the first i twin prime pairs.
%C A239309 a(2) = 0. Proof
%C A239309 It is easy to see that A054735(1)= 8 ==2 (mod 3) and A054735(n)==0 mod 3 for n > 1 where A054735 is the sum of twin pairs. Hence A086169(n)==2 (mod 3) and the prime 3 is never a divisor of A086169(n).
%H A239309 Michel Lagneau, <a href="/A239309/b239309.txt">Table of n, a(n) for n = 1..1000</a>
%e A239309 a(1)=1 because A086169(1)=(3+5)=8 and prime(1)= 2 divides 8;
%e A239309 a(2)=0 because prime(2)=3 is never a divisor of A086169(n);
%e A239309 a(3)=2 because A086169(2)=(3+5)+(5+7)=20 and prime(3)= 5 divides 20.
%t A239309 Transpose[With[{aprs=Thread[{Range[5000],Accumulate[Select[Table[Prime[n]+1,{n,45900}],PrimeQ[#+1]&]*2]}]},Flatten[Table[Select[aprs,Divisible[Last[#],Prime[m]]&,1],{m,1,60}],1]]][[1]]
%Y A239309 Cf. A000040, A054735, A086169.
%K A239309 nonn
%O A239309 1,3
%A A239309 _Michel Lagneau_, Mar 15 2014