A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Oct 01 2017

Keywords

• nonn
• look

Comments

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.

(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.

(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.

Compare this with the Redmond-Sun conjecture.

Examples

			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.
a(3) = 1 since A001597(3) = 8, A001597(4) = 9, 2*8^2 - 1 = 127 is prime but 2*9^2 - 1 is composite.
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.
a(14) = 1 since A001597(14) = 121, A001597(15) = 125, 2*125^2
- 1 = 31249 is prime but 2*k^2 - 1 is composite for every k = 121, 122, 123, 124.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• Wikipedia, Redmond-Sun conjecture

Crossrefs

Cf. A001597, A005153, A066049, A116086, A116455.

Programs

Mathematica

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}]