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A351373 a(n) is the least prime that begins a sequence of 2*n consecutive primes whose sum is 6 times a prime.

%I A351373 #9 Feb 22 2022 08:29:55
%S A351373 5,19,89,43,103,7,19,3,19,5,67,19,31,151,19,3,3,5,61,127,61,103,13,13,
%T A351373 137,109,149,67,59,103,59,271,983,31,3,43,277,181,3,683,307,307,83,
%U A351373 313,181,193,331,191,151,157,151,127,151,421,523,97,97,3,5,61,61,61,331,283,283,61,167,2003,263
%N A351373 a(n) is the least prime that begins a sequence of 2*n consecutive primes whose sum is 6 times a prime.
%C A351373 Conjecture: each odd prime occurs infinitely many times.
%H A351373 Robert Israel, <a href="/A351373/b351373.txt">Table of n, a(n) for n = 1..10000</a>
%e A351373 a(3) = 89 because the sum of the 6 consecutive primes starting with 89 is 89+97+101+103+107+109 = 606 = 6*101 where 101 is prime, and no smaller prime works.
%p A351373 P:= select(isprime,[seq(i,i=3..10^5,2)]):
%p A351373 S:= [0,op(ListTools:-PartialSums(P))]:
%p A351373 nP:= nops(S):
%p A351373 f:= proc(n) local i, s;
%p A351373   for i from 1 to nP-2*n do
%p A351373     s:= S[i+2*n]-S[i];
%p A351373     if s mod 6 = 0 and isprime(s/6) then return P[i] fi
%p A351373   od
%p A351373 end proc:
%p A351373 map(f, [$1..100]);
%Y A351373 Cf. A288632.
%K A351373 nonn
%O A351373 1,1
%A A351373 _J. M. Bergot_ and _Robert Israel_, Feb 09 2022