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A239328
Number of primes p < n with pi(p*n) - pi((p-1)n) prime, where pi(x) denotes the number of primes not exceeding x.
2
0, 0, 0, 1, 1, 2, 3, 2, 4, 3, 4, 4, 5, 5, 3, 4, 1, 2, 2, 5, 2, 2, 2, 5, 3, 3, 5, 8, 4, 7, 5, 5, 6, 5, 4, 4, 4, 6, 7, 5, 4, 6, 5, 8, 7, 4, 1, 2, 3, 8, 5, 3, 8, 5, 7, 6, 6, 4, 6, 9, 3, 6, 7, 8, 6, 3, 4, 4, 5, 6, 7, 6, 11, 10, 4, 6, 6, 9, 6, 5
OFFSET
1,6
COMMENTS
Conjecture: a(n) > 0 for all n > 3. Also, for any integer n > 4, there is a prime p < n with pi((p+1)*n) - pi(p*n) prime.
We have verified that a(n) > 0 for all n = 4, ..., 7*10^5.
LINKS
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
EXAMPLE
a(4) = 1 since 2 and pi(2*4) - pi(1*4) = 4 - 2 = 2 are both prime.
a(5) = 1 since 3 and pi(3*5) - pi(2*5) = 6 - 4 = 2 are both prime.
a(17) = 1 since 11 and pi(11*17) - pi(10*17) = 42 - 39 = 3 are both prime.
a(47) = 1 since 37 ad pi(37*47) - pi(36*47) = 270 - 263 = 7 are both prime.
MATHEMATICA
p[n_, k_]:=PrimeQ[PrimePi[Prime[k]*n]-PrimePi[(Prime[k]-1)*n]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 80}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 16 2014
STATUS
approved