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Search: id:A075540
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| A075540 |
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Integers that are the average of three successive primes. |
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6
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| 5, 53, 157, 173, 211, 257, 263, 373, 511, 537, 563, 593, 607, 653, 733, 947, 977, 999, 1073, 1103, 1123, 1187, 1223, 1239, 1367, 1461, 1501, 1511, 1541, 1747, 1753, 1763, 1773, 1899, 1907, 1917, 2071, 2181, 2287, 2401, 2409, 2417, 2449, 2677, 2903, 2963
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OFFSET
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1,1
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COMMENT
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Not every three successive primes have an integer average. The integer averages are in the sequence.
Not of these 3-averages are prime: the prime 3-averages are in A006562 (balanced primes). There are suprisingly many prime 3-averages: among first 117 3-averages, there are 59 primes. Indices i(n) of first prime in sequence of three primes with integer average are in A075541, for prime 3-averages i(n) are in A064113. Interprimes (s-averages with s=2) are all composite, see A024675.
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FORMULA
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a(n) = 1/3 (p(i)+p(i+1)+p(i+2)), for some i(n).
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EXAMPLE
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a(1) = 5 = 1/3(3+5+7), first integer average of three successive primes; next is: a(2) = 53 = 1/3(47 + 53 + 59); up to n=8 all a(n) are themselves prime; while a(9) = 511 = 1/3( 503 + 509 + 521) is first nonprime 3-average.
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MATHEMATICA
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A075540 = {}; Do[If[IntegerQ[s3 = (Prime[i] + Prime[i + 1] + Prime[i + 2])/3], A075540 = Append[A075540, s3]], {i, 1000}]; (*Length[A075540]=119*)
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CROSSREFS
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Cf. A006562, A024675, A075541, A064113.
Cf. A102655.
Sequence in context: A101190 A106097 A163580 this_sequence A006562 A094847 A001992
Adjacent sequences: A075537 A075538 A075539 this_sequence A075541 A075542 A075543
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KEYWORD
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nonn
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AUTHOR
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Zak Seidov (zakseidov(AT)yahoo.com), Sep 21 2002
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