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A218114 Integer arithmetic means of 10 consecutive primes.

%I A218114 #30 Apr 24 2023 12:30:32
%S A218114 30,34,51,78,87,106,132,165,180,203,225,231,248,253,305,312,375,381,
%T A218114 488,502,510,588,633,690,727,734,754,761,988,1038,1070,1089,1110,1140,
%U A218114 1183,1218,1299,1336,1368,1408,1416,1431,1437,1449,1466,1504,1525,1600,1632
%N A218114 Integer arithmetic means of 10 consecutive primes.
%C A218114 It is obvious that the terms occur in increasing order, since the mean increases by (prime(n)-prime(n-10))/10 when going to the 10 primes which include prime(n) as largest term. However, differences of 6, as e.g. between the terms a(n)=9813497 and a(n+1)=9813503 (= average of prime(653096) through prime(653105)), seem to occur infinitely often. Is this true, and is this the smallest such step? - _M. F. Hasler_, Oct 21 2012
%C A218114 Also difference 5 seems to occur infinitely often. For first 200000 differences, values 5..10 occur 5, 57, 123, 400, 1755, 1439 times. Apparently all differences >4 occur infinitely often. - _Zak Seidov_, May 22 2015
%H A218114 Zak Seidov, <a href="/A218114/b218114.txt">Table of n, a(n) for n = 1..1000</a>
%e A218114 a(1) is derived from (prime(6)+...+prime(15))/10 = (13+ 17+ 19+ 23+ 29+ 31+ 37+ 41+ 43+ 47)/10=30.
%p A218114 Psums:= ListTools:-PartialSums(select(isprime,[2,(2*i+1 $ i=1..10^4)])):
%p A218114 select(type, (Psums[11..-1] - Psums[1..-11])/10, integer); # _Robert Israel_, May 22 2015
%t A218114 Select[Total /@ Partition[Prime@ Range@ 263, 10, 1]/10, IntegerQ] (* _Michael De Vlieger_, May 22 2015 *)
%t A218114 Select[Mean/@Partition[Prime[Range[300]],10,1],IntegerQ] (* _Harvey P. Dale_, Aug 28 2021 *)
%o A218114 (PARI) lista(nn) = {for (n=1, nn, my(s = sum(k=0, 9, prime(n+k))/10); if (type(s) == "t_INT", print1(s, ", ")););} \\ _Michel Marcus_, May 23 2015
%Y A218114 Cf. A000040, A123096 (subsequence of primes), A127337.
%K A218114 nonn
%O A218114 1,1
%A A218114 _Zak Seidov_, Oct 21 2012