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A218114 Integer arithmetical averages of 10 consecutive primes. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes are listed in A123096 = (727, 761, 1663, 1783, 1811, ...).

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

LINKS

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000040, A123096, A127337.

Sequence in context: A095992 A061842 A109226 * A138689 A257447 A257440

Adjacent sequences:  A218111 A218112 A218113 * A218115 A218116 A218117

KEYWORD

nonn

AUTHOR

Zak Seidov, Oct 21 2012

STATUS

approved

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Last modified October 5 06:51 EDT 2022. Contains 357252 sequences. (Running on oeis4.)