

A111287


a(n) = smallest k such that prime(n) divides Sum_{i=1..k} prime(i).


8



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, 422, 240, 24, 94, 22, 16, 173, 861, 231, 143, 140, 213, 902, 18, 134, 143, 310, 70, 58, 12, 550, 237, 210, 229, 57, 221, 271, 194, 540, 145, 718, 116, 184, 90, 14, 168, 455, 61, 454
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OFFSET

1,2


COMMENTS

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.
To clarify the previous comment: If the sum of the primes up to some point is s (mod b), then we need exactly bs consecutive primes equal to 1 (mod b) to produce a sum divisible by b. Hence when there are b1 consecutive primes congruent to 1 (mod b), then the sum of primes up to one of those primes will be divisible by b.  T. D. Noe, Dec 02 2009


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359373 [MR1760689]


EXAMPLE

A007504 begins 2,5,10,17,28,41,58,77,100,129,... and the k=10th term is the first one that is divisible by prime(2) = 3, so a(2) = 10 (see also A103208).


MAPLE

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;


MATHEMATICA

Table[p=Prime[n]; s=0; k=0; While[k++; s=Mod[s+Prime[k], p]; s>0]; k, {n, 10}]  T. D. Noe, Dec 02 2009


PROG

(PARI) A111287(n)= n=Mod(0, prime(n)); for(k=1, 1e9, (n+=prime(k))  return(k)) \\ M. F. Hasler, Nov 29 2009


CROSSREFS

Cf. A000041, A007504, A053050, A111267, A111272, A103208, A168678.
Sequence in context: A069036 A155817 A037922 * A255668 A187815 A274606
Adjacent sequences: A111284 A111285 A111286 * A111288 A111289 A111290


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 03 2005


EXTENSIONS

The comments are based on correspondence with Paul Pollack and a posting to sci.math by Fred Helenius.
Typo in reference fixed by David Applegate, Dec 18 2009


STATUS

approved



