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A254687
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Number of decompositions of 2n into sums of two primes p1 < p2 such that p2-p1-1 is also a prime.
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2
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0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 0, 1, 2, 3, 3, 1, 2, 3, 3, 3, 2, 2, 1, 1, 4, 2, 2, 3, 2, 4, 3, 3, 3, 2, 3, 4, 3, 4, 2, 4, 2, 2, 3, 2, 5, 3, 2, 4, 5, 5, 5, 4, 4, 1, 4, 5, 2, 4, 2, 4, 3, 3, 4, 4, 2, 5, 3, 5, 1, 5, 3, 0, 6, 4, 5, 4, 2, 6, 4, 5
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OFFSET
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1,9
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COMMENTS
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a(n)=0 for n = 1, 2, 3, 4, 6, 18, 79. It is conjectured that there is not any other n for which a(n)=0.
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LINKS
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EXAMPLE
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n=5, 2n=10=3+7. 7-3-1=3 is prime, so a(5)=1;
n=6, 2n=12=5+7. 7-5-1=1 is not prime, so a(6)=0;
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n=21, 2n=42=5+37=11+31=13+29=19+23. 37-5-1=31 is prime, 31-11-1=19 is prime, 29-13-1=15 is composite, 23-19-1=3 is prime: three primes in the form of p2-p1-1 found, so a(21)=3.
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MATHEMATICA
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Table[e = 2 n; ct = 0; p1 = 1; While[p1 = NextPrime[p1]; p1 < n, p2 = e - p1; If[PrimeQ[p2], If[PrimeQ[Abs[p2 - p1 - 1]], ct++]]]; ct, {n, 1, 100}]
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PROG
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(Python)
from sympy import isprime, nextprime
....y, x, n2 = 0, 2, 2*n
....while x < n:
........if isprime(n2-x) and isprime(n2-2*x-1):
............y += 1
........x = nextprime(x)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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