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A238277
a(n) = |{0 <= k < n: the number of primes in the interval (k*n, (k+1)*n] is a square}|.
6
1, 2, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 1, 5, 3, 3, 10, 11, 8, 7, 10, 6, 13, 11, 13, 8, 12, 10, 8, 7, 7, 6, 4, 5, 5, 6, 3, 4, 7, 3, 7, 7, 8, 7, 7, 9, 8, 12, 8, 5, 12, 11, 14, 11, 14, 11, 8, 11, 9, 9, 13, 12, 5, 14, 15, 12, 15, 12, 15, 14, 15, 16, 13, 10, 18, 20, 12, 7, 17, 13
OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
We have verified this for n up to 10^5.
See also A238278 and A238281 for related conjectures.
LINKS
EXAMPLE
a(9) = 1 since the interval (0, 9] contains exactly 2^2 = 4 primes.
a(13) = 1 since the interval (9*13, 10*13] contains exactly 1^2 = 1 prime.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
d[k_, n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
a[n_]:=Sum[If[SQ[d[k, n]], 1, 0], {k, 0, n-1}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 22 2014
STATUS
approved