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%I #14 Apr 06 2014 22:29:33
%S 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,
%T 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,
%U 13,12,5,14,15,12,15,12,15,14,15,16,13,10,18,20,12,7,17,13
%N a(n) = |{0 <= k < n: the number of primes in the interval (k*n, (k+1)*n] is a square}|.
%C Conjecture: a(n) > 0 for all n > 0.
%C We have verified this for n up to 10^5.
%C See also A238278 and A238281 for related conjectures.
%H Zhi-Wei Sun, <a href="/A238277/b238277.txt">Table of n, a(n) for n = 1..3000</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(9) = 1 since the interval (0, 9] contains exactly 2^2 = 4 primes.
%e a(13) = 1 since the interval (9*13, 10*13] contains exactly 1^2 = 1 prime.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t d[k_,n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
%t a[n_]:=Sum[If[SQ[d[k,n]],1,0],{k,0,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000290, A237598, A237706, A238278, A238281.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Feb 22 2014
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