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A300013
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a(n) is the number of primes p such that both 2n-p and 2n+2-nextprime(p) are prime numbers.
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0
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0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 6, 3, 3, 5, 5, 4, 6, 7, 5, 4, 5, 4, 6, 4, 4, 9, 3, 3, 9, 8, 5, 7, 8, 5, 6, 8, 5, 7, 7, 3, 8, 4, 3, 10, 9, 4, 8, 9, 8, 10, 10, 7, 10, 7, 5, 9, 5, 4, 12, 10, 3, 7, 9, 8, 12, 11, 5, 10, 6, 7, 15, 9, 6, 11, 9, 3, 10
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OFFSET
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1,4
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COMMENTS
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In the name, "nextprime(p)" stands for the smallest prime number that is greater than p.
Conjecture: a(n) > 0 for all integer n > 1.
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LINKS
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Table of n, a(n) for n=1..83.
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EXAMPLE
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For n=2, 2n=4, 2n+2=6. Both 4-2=2 and 6-nextprime(2)=6-3=3 are primes. This is the only case, so a(2)=1;
For n=3, 2n=6, 2n+2=8. Both 6-3=5 and 8-nextprime(3)=8-5=3 are primes. This is the only case, so a(3)=1;
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For n=8, 2n=16, 2n+2=18. The following cases satisfy the definition:
1) 16-3=13, 18-nextprime(3)=18-5=13;
2) 16-5=11, 18-nextprime(5)=18-7=11;
3) 16-11=5, 18-nextprime(11)=18-13=5.
So a(8)=3;
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For n=10, 2n=20, 2n+2=22. The following cases satisfy the definition:
1) 20-3=17, 22-nextprime(3)=22-5=17;
2) 20-7=13, 22-nextprime(7)=22-11=11;
3) 20-13=7, 22-nextprime(13)=22-17=5;
4) 20-17=3, 22-nextprime(17)=22-19=3.
So a(10)=4.
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MATHEMATICA
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Table[n = i*2; np2 = n + 2; p = 1; ct = 0; While[p = NextPrime[p]; p < n, If[PrimeQ[n - p] && (cp = np2 - NextPrime[p]; (cp > 0) && PrimeQ[cp]), ct++]]; ct, {i, 1, 83}]
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PROG
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(PARI) a(n) = sum(k=1, primepi(2*n), isprime(2*n-prime(k)) && isprime(2*n+2-prime(k+1))); \\ Michel Marcus, Jun 21 2018
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CROSSREFS
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Cf. A000040, A002375, A045917, A002372.
Sequence in context: A116513 A122651 A343378 * A130535 A329194 A210533
Adjacent sequences: A300010 A300011 A300012 * A300014 A300015 A300016
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KEYWORD
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nonn,easy
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AUTHOR
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Lei Zhou, Jun 18 2018
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STATUS
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approved
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