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A334810
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The number of even numbers between 4 and 2p that cannot be written as the sum of two primes less than or equal to the n-th prime number p.
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3
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0, 0, 0, 0, 1, 0, 1, 0, 1, 4, 1, 3, 2, 1, 1, 4, 7, 2, 4, 3, 1, 4, 3, 4, 6, 5, 2, 2, 0, 1, 8, 7, 8, 3, 8, 5, 5, 7, 5, 6, 7, 2, 8, 4, 3, 1, 7, 14, 10, 7, 4, 6, 3, 7, 8, 11, 14, 8, 6, 5, 3, 8, 14, 10, 7, 6, 11, 13, 15, 10, 7, 7, 9, 10, 11, 8, 9, 10, 7, 9, 13, 9, 13, 10, 9, 6, 6, 8, 7, 3, 2, 9, 10, 10, 10, 10, 8, 15, 9, 20
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OFFSET
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1,10
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COMMENTS
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This sequence is related to the Goldbach conjecture: any even number (m) greater than 4 can be written as the sum of two odd primes (i.e., m = p1 + p2). For any given prime number p, if the restriction (p1, p2 <= p) is applied, some even numbers less than 2p may not be written as the sum of two prime numbers. The prime numbers corresponding to a(n)=0 in this sequence are the seven prime numbers listed in A301776.
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LINKS
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EXAMPLE
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a(1)=0. The 1st prime is 2. Even number 4 can be written as 2+2.
a(2)=0. The 2nd prime is 3. Even numbers 4 and 6 can be written as: 4=2+2 and 6=3+3.
a(3)=0. The 3rd prime is 5. Even numbers between 4 and 10 are: 4=2+2, 6=3+3, 8=3+5, and 10=5+5.
a(5)=1. The 5th prime is 11. Among the 10 even numbers between 4 and 22, only 20 cannot be written as the sum of two primes <= 11.
a(10)=4. The 10th prime is 29. Four even numbers (44, 50, 54, and 56) between 4 and 58 cannot be written as the sum of two primes <= 29.
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MATHEMATICA
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a[n_] := Block[{p = Prime[n], r = Prime@ Range@ n}, Sum[Boole[{} == IntegerPartitions[2 k, {2}, r]], {k, 2, p}]]; Array[a, 83] (* Giovanni Resta, May 12 2020 *)
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CROSSREFS
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Cf. A301776 (prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p).
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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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