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%I #90 Aug 05 2019 09:26:52
%S 0,1,1,1,2,2,1,1,3,3,2,2,1,2,4,4,4,5,3,2,4,1,2,1,4,4,4,2,3,4,4,4,3,2,
%T 5,4,4,4,3,5,4,5,3,4,7,6,5,2,5,3,5,7,1,3,5,5,4,6,5,4,4,5,3,1,4,7,6,5,
%U 5,4,5,7,4,5,3,5,6,8,3,4,4,6,3,5,2,2,3,6,6,4,5,6,5,5,8,6,4,7,5,4
%N Number of ways to write n = x + y (x, y > 0) with 2^x + y prime.
%C Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 7 can be written as x + y with 0 < x < y such that 2^x + y is prime.
%C (ii) Every n = 2, 3, ... can be written as x + y (x, y > 0) with 2^x + y*(y+1)/2 prime.
%C (iii) Each integer n > 1 can be written as x + y (x, y > 0) with 2^x + y^2 - 1 prime. Also, any integer n > 1 not equal to 16 can be written as x + y (x, y > 0) with 2^x + y^4 - 1 prime.
%C We have verified part (i) of the conjecture for n up to 1.6*10^6. For example, 421801 = 149536 + 272265 with 2^149536 + 272265 prime.
%C We have extended our verification of part (i) of the conjecture for n up to 2*10^6. For example, 1657977 = 205494 + 1452483 with 2^205494 + 1452483 prime. - _Zhi-Wei Sun_, Aug 30 2015
%C The verification of part (i) of the conjecture has been made for n up to 7.29*10^6. For example, we find that 5120132 = 250851 + 4869281 with 2^250851 + 4869281 a prime of 75514 decimal digits. - _Zhi-Wei Sun_, Nov 16 2015
%C We have finished the verification of part (i) of the conjecture for n up to 10^7. For example, we find that 9302003 = 311468 + 8990535 with 2^311468 + 8990535 a prime of 93762 decimal digits. - _Zhi-Wei Sun_, Jul 28 2016
%C In a paper published in 2017, the author announced a USD $1000 prize for the first solution to his conjecture that a(n) > 0 for all n > 1. - _Zhi-Wei Sun_, Dec 03 2017
%H Zhi-Wei Sun, <a href="/A231201/b231201.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;85566cc1.1311">Write n = k + m with 2^k + m prime</a>, a message to Number Theory List, Nov. 16, 2013.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv:1312.1166 [math.NT], 2013-2014.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2015.
%H Z.-W. Sun, <a href="http://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>.)
%e a(8) = 1 since 8 = 3 + 5 with 2^3 + 5 = 13 prime.
%e a(53) = 1 since 53 = 20 + 33 with 2^{20} + 33 = 1048609 prime.
%e a(64) = 1 since 64 = 13 + 51 with 2^{13} + 51 = 8243 prime.
%t a[n_]:=Sum[If[PrimeQ[2^x+n-x],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000079, A228425, A228428, A228429, A228430, A228431, A231516, A231555, A231557, A231561, A231577, A231631.
%K nonn,look
%O 1,5
%A _Zhi-Wei Sun_, Nov 10 2013
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