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A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.

%I A293395 #31 Jul 31 2025 16:25:55
%S A293395 71,271,337,431,631,661,769,1153,1721,1789,2131,2339,2381,2749,2777,
%T A293395 3313,3319,3517,3919,4139,4337,4729,4789,4903,4937,4993,5171,5303,
%U A293395 5323,5507,5849,5851,6271,6323,6451,6959,6983,7489,7919,8221,8363,8419,9349,9613,9619
%N A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.
%C A293395 3313 is the smallest term such that 3313 +- 6 are both prime.
%H A293395 Charles R Greathouse IV, <a href="/A293395/b293395.txt">Table of n, a(n) for n = 1..10000</a>
%e A293395 71 is a term because it is the initial member of 5 consecutive primes {71, 73, 79, 83, 89} and (71 + 73 + 79 + 83 + 89)/5 = 79.
%e A293395 271 is a term because it is the initial member of 5 consecutive primes {271, 277, 281, 283, 293} and (271 + 277 + 281 + 283 + 293)/5 = 281.
%p A293395 A293395:= proc(n)local a, b, c, d, e; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); d:=ithprime(n+3); e:=ithprime(n+4); if (a + b + d + e)/4 = c then RETURN (a); fi; end: seq(A293395(n), n=1..3000);
%t A293395 Select[Prime@ Range[1200], #[[3]] == Mean@ Delete[#, 3] &@ NestList[NextPrime, #, 4] &] (* _Michael De Vlieger_, Oct 09 2017 *)
%t A293395 Select[Partition[Prime[Range[1200]],5,1],Mean[#]==#[[3]]&][[;;,1]] (* _Harvey P. Dale_, Jul 31 2025 *)
%o A293395 (PARI) for(n=1, 1000, a=prime(n); b=prime(n+1); c=prime(n+2); d=prime(n+3); e=prime(n+4); if((a+b+d+e)/4==c, print1(a,", ")));
%o A293395 (PARI) list(lim)=my(v=List(),p=2,q=3,r=5,s=7); forprime(t=11,lim, if(p+q+s+t==4*r, listput(v,p)); p=q; q=r; r=s; s=t); Vec(v) \\ _Charles R Greathouse IV_, Oct 09 2017
%Y A293395 Cf. A006562, A034964, A122535.
%K A293395 nonn
%O A293395 1,1
%A A293395 _K. D. Bajpai_, Oct 08 2017
%E A293395 Definiyion simplified by _David A. Corneth_, Oct 14 2017
%E A293395 Examples clarified by _Harvey P. Dale_, Jul 31 2025