A328264 a(n) is the least prime p such that prime(n) divides the sum of n consecutive primes starting with p.

2, 5, 2, 37, 83, 17, 7, 23, 13, 67, 163, 821, 227, 7, 13, 151, 599, 643, 271, 2, 83, 19, 83, 1069, 61, 37, 823, 263, 23, 857, 89, 1931, 139, 181, 71, 239, 1861, 739, 487, 37, 1237, 3833, 37, 6961, 1709, 499, 587, 271, 2687, 359, 5, 727, 73, 491, 73, 41, 3989, 797, 2083, 1451, 199, 349, 2027, 2441

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 09 2019

Keywords

• nonn

Comments

a(n)=3 for n=850 and 55154 (and presumably infinitely many others).

Examples

			a(4)=37 because prime(4)=7 divides the sum of 4 consecutive primes starting with 37 (37+41+43+47=168), but does not divide any earlier sum of 4 consecutive primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..9332

Crossrefs

Cf. A024011 (a(n)=2).

Programs

Maple

P:= [0,seq(ithprime(i),i=1..100000)]:
S:= ListTools:-PartialSums(P):
f:= proc(n) local p,k;
  p:= ithprime(n);
  for k from 1 to nops(S)-n do
      if S[k+n]-S[k] mod p = 0 then
      return P[k+1]
      fi
    od;
  FAIL
end proc:
map(f, [$1..200]);

Mathematica

a[n_] := Block[{m=Prime@n, s=Sum[Prime@i, {i, n}], p=2, q}, q=m; While[Mod[s, m] > 0, s-=p; {p, q} = NextPrime@{p, q}; s+=q]; p]; Array[a, 70] (* Giovanni Resta, Oct 10 2019 *)