%I A111287
%S A111287 1,10,2,5,8,49,4,23,23,7,39,29,6,10,39,25,30,151,38,19,139,27,174,21,287,
%T A111287 422,240,24,94,22,16,173,861,231,143,140,213,902,18,134,143,310,70,58,
               12,
%U A111287 550,237,210,229,57,221,271,194,540,145,718,116,184,90,14,168,455,61,454
%N A111287 a(n) = smallest k such that prime(n) divides Sum_{i=1..k} prime(i).
%C A111287 It follows from a theorem of Daniel Shiu that k always exists. Shiu has 
               proved that if (a,b) = 1 then the arithmetic progression a, a + b, 
               ..., a + k*b, ... contains arbitrarily long sequences of consecutive 
               primes. Since, for any positive integer b, there are thus arbitrarily 
               long sequences of consecutive primes congruent to 1 mod b, there 
               must be infinitely many a(n) that are divisible by b.
%C A111287 To clarify the previous comment: If the sum of the primes up to some 
               point is s (mod b), then we need exactly b-s consecutive primes equal 
               to 1 (mod b) to produce a sum divisible by b. Hence when there are 
               b-1 consecutive primes congruent to 1 (mod b), then the sum of primes 
               up to one of those primes will be divisible by b. [From T. D. Noe 
               (noe(AT)sspectra.com), Dec 02 2009]
%D A111287 D. K. L. Shiu, Strings of congruent primes, J. London Math. Soc. 61 (2000), 
               359-373; MR 2001f:11155.
%H A111287 T. D. Noe, <a href="b111287.txt">Table of n, a(n) for n=1..10000</a>
%H A111287 D. K. L. Shiu, <a href="http://dx.doi.org/10.1112/S0024610799007863">
               Strings of Congruent Primes</a>, J. Lond. Math. Soc. 61 (2) (2000) 
               359-373 [<a href=""http://www.ams.org/mathscinet-getitem?mr=1760689">
               MR1760689</a>] [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 30 2009]
%e A111287 A007504 begins 2,5,10,17,28,41,58,77,100,129,... and the k=10-th term 
               is the first one that is divisible by prime(2) = 3, so a(2) = 10 
               (see also A103208).
%p A111287 read transforms; M:=1000; p0:=[seq(ithprime(i),i=1..M)]; q0:=PSUM(p0); 
               w:=[]; for n from 1 to M do p:=p0[n]; hit := 0; for i from 1 to M 
               do if q0[i] mod p = 0 then w:=[op(w),i]; hit:=1; break; fi; od: if 
               hit = 0 then break; fi; od: w;
%t A111287 Table[p=Prime[n]; s=0; k=0; While[k++; s=Mod[s+Prime[k],p]; s>0]; k, 
               {n,10}] [From T. D. Noe (noe(AT)sspectra.com), Dec 02 2009]
%o A111287 (PARI) A111287(n)={ n=Mod(0,prime(n)); for(k=1,1e9, (n+=prime(k))|return(k))} 
               \\ [From M. F. Hasler (mhasler(AT)univ-ag.fr), Nov 29 2009]
%Y A111287 Cf. A000041, A007504, A053050, A111267, A111272, A103208, etc.
%Y A111287 Sequence in context: A069036 A155817 A037922 this_sequence A084455 A069532 
               A084461
%Y A111287 Adjacent sequences: A111284 A111285 A111286 this_sequence A111288 A111289 
               A111290
%Y A111287 Cf. A168678 [From T. D. Noe (noe(AT)sspectra.com), Dec 02 2009]
%K A111287 nonn,new
%O A111287 1,2
%A A111287 N. J. A. Sloane (njas(AT)research.att.com), Nov 03 2005
%E A111287 The comments are based on correspondence with Paul Pollack and a posting 
               to sci.math by Fred Helenius.