OFFSET
1,3
COMMENTS
Conjecture: (i) a(n) is at least sqrt(n-1) for each n > 0, and equality holds only when n is 2 or 26.
(ii) The sequence contains all positive integers.
We have verified part (i) of the conjecture for n up to 10000.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014
EXAMPLE
a(1) = 1 since the interval (0,1*1] contains no prime, and the set {0} has cardinaly 1.
a(3) = 2 since the intervals (0, 1*3], (1*3, 2*3], (2*3, 3*3] contain exactly 2, 1, 1 primes respectively, and the set {2, 1, 1} has cardinality 2.
MATHEMATICA
d[k_, n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
a[n_]:=Length[Union[Table[d[k, n], {k, 0, n-1}]]]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 23 2014
STATUS
approved