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A218114
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Integer arithmetic means of 10 consecutive primes.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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a(1) is derived from (prime(6)+...+prime(15))/10 = (13+ 17+ 19+ 23+ 29+ 31+ 37+ 41+ 43+ 47)/10=30.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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