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.
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
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Jun 18 2018
STATUS
approved