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A382699 First member of the least set of 4 consecutive primes such that the sum of each pair of consecutive primes in this set is a multiple of n.

%I A382699 #20 Apr 19 2025 18:10:12
%S A382699 2,3,5,23,157,5,977,53,5171,157,33871,137,159293,977,2969,541,406873,
%T A382699 5171,471313,6047,166739,33871,2112193,5309,520763,159293,207869,5443,
%U A382699 2404471,2969,1531487,88919,2673791,406873,6056569,95737,8480357,471313,561829,73477
%N A382699 First member of the least set of 4 consecutive primes such that the sum of each pair of consecutive primes in this set is a multiple of n.
%H A382699 Paolo P. Lava, <a href="/A382699/b382699.txt">Table of n, a(n) for n = 1..100</a>
%H A382699 Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_092.htm">Conjecture 92. For any integer m there is at least one set of consecutive primes...</a>, The Prime Puzzles and Problems Connection.
%e A382699 a(4) = 23. The least 4 consecutive primes are 23, 29, 31, 37:
%e A382699   23 + 29 = 52 and 52/4 = 13;
%e A382699   29 + 31 = 60 and 60/4 = 15;
%e A382699   31 + 37 = 68 and 68/4 = 17.
%e A382699 a(37) = 8480357. The least 4 consecutive primes are 8480357, 8480369, 8480431, 8480443:
%e A382699   8480357 + 8480369 = 16960726 and 16960726/37 = 458398;
%e A382699   8480369 + 8480431 = 16960800 and 16960800/37 = 458400;
%e A382699   8480431 + 8480443 = 16960874 and 16960874/37 = 458402.
%p A382699 P:=proc(q) local a,b,c,d,n,v; v:=[];for n from 1 to 30 do a:=2; b:=3; c:=5; d:=7;
%p A382699 while true do if frac((a+b)/n)=0 and frac((b+c)/n)=0 and frac((c+d)/n)=0 then v:=[op(v),a]; break;
%p A382699 else a:=b; b:=c; c:=d; d:=nextprime(d); fi; od; od; op(v); end: P(2*10^6);
%t A382699 Do[p=0;Until[Mod[Prime[p]+Prime[p+1],n]==0&&Mod[Prime[p+1]+Prime[p+2],n]==0&&Mod[Prime[p+2]+Prime[p+3],n]==0,p++];a[n]=Prime[p],{n,45}];Array[a,40] (* _James C. McMahon_, Apr 09 2025 *)
%Y A382699 Cf. A254862 (2 consecutive), A382698 (3 consecutive), A382700 (5 consecutive).
%K A382699 nonn
%O A382699 1,1
%A A382699 _Paolo P. Lava_, Apr 04 2025