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A300013 a(n) is the number of primes p such that both 2n-p and 2n+2-nextprime(p) are prime numbers. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

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;

...

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;

...

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.

MATHEMATICA

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}]

PROG

(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

CROSSREFS

Cf. A000040, A002375, A045917, A002372.

Sequence in context: A116513 A122651 A343378 * A130535 A329194 A210533

Adjacent sequences:  A300010 A300011 A300012 * A300014 A300015 A300016

KEYWORD

nonn,easy

AUTHOR

Lei Zhou, Jun 18 2018

STATUS

approved

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Last modified September 18 14:52 EDT 2021. Contains 347527 sequences. (Running on oeis4.)