OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 for no n > 144. Moreover, for any positive integer n, there is a prime p < sqrt(2*n)*log(5n) with p*n + pi(p) prime.
(ii) For each integer n > 8, there is a prime p <= n + 1 with (p-1)*n - pi(p-1) prime.
(iii) For every n = 1, 2, 3, ... there is a positive integer k < 3*sqrt(n) with k*n + prime(k) prime.
(iv) For each n > 13, there is a positive integer k < n with k*n + prime(n-k) prime.
We have verified that a(n) > 0 for all n = 5, ..., 10^8.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014
EXAMPLE
a(3) = 1 since 2 and 2*3 + pi(2) = 6 + 1 = 7 are both prime.
a(10) = 1 since 5 and 5*10 + pi(5) = 50 + 3 = 53 are both prime.
a(107) = 1 since 89 and 89*107 + pi(89) = 9523 + 24 = 9547 are both prime.
a(144) = 1 since 59 and 59*144 + pi(59) = 8496 + 17 = 8513 are both prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[Prime[k]*n+k], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 80}]
PROG
(PARI) vector(100, n, sum(k=1, primepi(n-1), isprime(prime(k)*n+k))) \\ Colin Barker, Feb 08 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 08 2014
STATUS
approved