A123096 -id:A123096 - cp's OEIS frontend

A218114 Integer arithmetic means of 10 consecutive primes.

30, 34, 51, 78, 87, 106, 132, 165, 180, 203, 225, 231, 248, 253, 305, 312, 375, 381, 488, 502, 510, 588, 633, 690, 727, 734, 754, 761, 988, 1038, 1070, 1089, 1110, 1140, 1183, 1218, 1299, 1336, 1368, 1408, 1416, 1431, 1437, 1449, 1466, 1504, 1525, 1600, 1632

Offset: 1

Zak Seidov, Oct 21 2012

nonn

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

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

			a(1) is derived from (prime(6)+...+prime(15))/10 = (13+ 17+ 19+ 23+ 29+ 31+ 37+ 41+ 43+ 47)/10=30.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A123096 (subsequence of primes), A127337.

Psums:= ListTools:-PartialSums(select(isprime,[2,(2*i+1 $ i=1..10^4)])):
select(type, (Psums[11..-1] - Psums[1..-11])/10, integer); # Robert Israel, May 22 2015

Select[Total /@ Partition[Prime@ Range@ 263, 10, 1]/10, IntegerQ] (* Michael De Vlieger, May 22 2015 *)
Select[Mean/@Partition[Prime[Range[300]],10,1],IntegerQ] (* Harvey P. Dale, Aug 28 2021 *)

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

A217985 Primes that are arithmetical average of 100 consecutive primes.

Original entry on oeis.org

5009, 6047, 8039, 9311, 9497, 11171, 15137, 17029, 18869, 20983, 26339, 28627, 31699, 35023, 38833, 43579, 49199, 49727, 50549, 60089, 68447, 72469, 76603, 87931, 89659, 98809, 99733, 102547, 111217, 143281, 143831, 150431, 185401, 191341, 195137, 195161

Offset: 1

Zak Seidov, Oct 21 2012

nonn

			a(1) = 5009 because  prime(620)+...+prime(719) = 4583+...+5441 = 500900.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A123096, A127337.

h = 100; t = {}; s = Prime[Range[h]]; pCnt = h; Do[sm = Total[s]; If[Mod[sm, h] == 0 && PrimeQ[sm/h], AppendTo[t, sm/h]]; pCnt++; s = Append[Rest[s], Prime[pCnt]], {20000}]; t (* T. D. Noe, Oct 22 2012 *)

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A218114 Integer arithmetic means of 10 consecutive primes.

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