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%I #13 Oct 03 2017 20:52:37
%S 3,4,1,4,6,1,1,2,9,8,6,7,7,1,3,6,8,11,8,1,6,5,11,14,4,2,12,14,16,8,6,
%T 15,13,9,16,16,15,15,13,10,6,16,21,16,11,4,8,22,23,17,20,7,8,23,18,21,
%U 4,23,13,1,4,24,28,24,24,24,8,14,23,24,25,1,24,15,2,21,29,26,24,35,27,25,31,30,31,30,24,4,30,30,32,30,35,31,13,13,33,31,29,31
%N a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.
%C Conjecture: (i) a(n) > 0 for all n > 0. In other words, for any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z^2 - 1 is prime.
%C (ii) For any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z + 3 (or 20*z^2 + 3) is prime.
%C (iii) For perfect powers x^m and y^n with 0 < x^m < y^n, there is a practical number q (cf. A005153) with x^m <= q <= y^n, unless x^m = 5^2 and y^n = 3^3, or x^m = 11^2 and y^n = 5^3, or x^m = 22434^2 and y^n = 55^5.
%C Compare this with the Redmond-Sun conjecture.
%H Zhi-Wei Sun, <a href="/A293190/b293190.txt">Table of n, a(n) for n = 1..5000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Redmond%E2%80%93Sun_conjecture">Redmond-Sun conjecture</a>
%e a(1) = 3 since 2*2^2 - 1, 2*3^2-1 and 2*4^2-1 are all prime but 2*1^2 - 1 is not prime.
%e a(3) = 1 since A001597(3) = 8, A001597(4) = 9, 2*8^2 - 1 = 127 is prime but 2*9^2 - 1 is composite.
%e a(6) = 1 since A001597(6) = 25, A001597(7) = 27, 2*25^2 - 1 = 1249 is prime but 2*26^2 - 1 and 2*27^2 - 1 are composite.
%e a(14) = 1 since A001597(14) = 121, A001597(15) = 125, 2*125^2
%e - 1 = 31249 is prime but 2*k^2 - 1 is composite for every k = 121, 122, 123, 124.
%e a(361) = 1 since A001597(361) = 46^3 = 97336, A001597(362) = 312^2 = 97344, and k = 97342 is the only number among 97336,...,97344 with 2*k^2 - 1 prime.
%t n=1;m=1;Do[Do[If[IntegerQ[k^(1/Prime[i])],Print[n," ",Sum[Boole[PrimeQ[2j^2-1]],{j,m,k}]];n=n+1;m=k;Goto[aa]],{i,1,PrimePi[Log[2,k]]}];Label[aa],{k,2,6561}]
%Y Cf. A001597, A005153, A066049, A116086, A116455.
%K nonn,look
%O 1,1
%A _Zhi-Wei Sun_, Oct 01 2017
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