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%I #26 Jul 10 2015 20:05:27
%S 0,1,2,1,2,2,1,2,2,5,2,3,3,1,6,5,3,3,1,4,2,4,4,3,4,2,4,3,1,4,3,3,7,5,
%T 4,5,5,4,3,2,5,2,2,4,5,4,9,7,4,3,2,4,3,4,3,2,4,6,5,6,4,4,2,2,7,5,6,6,
%U 8,3,7,3,5,6,10,6,6,6,4,5
%N Number of odd primes p < 2*n such that the number of squarefree numbers among 1, ..., ((p-1)/2)*n is prime.
%C Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 4, 7, 14, 19, 29.
%C This is an analog of the conjecture in A237578 for squarefree numbers. We have verified it for n up to 20000.
%C See also A238646 for a similar conjecture involving squarefree numbers.
%H Zhi-Wei Sun, <a href="/A238645/b238645.txt">Table of n, a(n) for n = 1..900</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e a(4) = 1 since 3 is prime and there are exactly 3 squarefree numbers among 1, ..., (3-1)/2*4 (namely, 1, 2, 3).
%e a(14) = 1 since 5 and 17 are both prime, and there are exactly 17 squarefree numbers among 1, ..., (5-1)/2*14.
%e a(19) = 1 since 3 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., (3-1)/2*19 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).
%e a(29) = 1 since 41 and 353 are both prime, and there are exactly 353 squarefree numbers among 1, ..., (41-1)/2*29 = 580.
%t s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]
%t a[n_]:=Sum[If[PrimeQ[s[(Prime[k]-1)/2*n]],1,0],{k,2,PrimePi[2n-1]}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A005117, A013928, A237578, A237615, A238646.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Mar 02 2014
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