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A230022
a(n) = |{the number of primes in the interval (k*n, (k+1)*n]: k = 0, 1, ..., n-1}|.
1
1, 1, 2, 2, 3, 3, 3, 4, 3, 4, 5, 5, 4, 5, 4, 5, 5, 6, 7, 6, 5, 6, 6, 6, 5, 5, 7, 6, 6, 7, 7, 6, 6, 7, 7, 8, 9, 8, 9, 9, 8, 8, 8, 9, 8, 9, 9, 8, 10, 10, 9, 10, 9, 10, 10, 10, 10, 11, 10, 10, 9, 10, 9, 11, 10, 11, 11, 11, 11, 11, 11, 11, 10, 12, 11, 10, 11, 12, 13, 11
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
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