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%I #17 Aug 06 2019 17:39:47
%S 0,0,1,2,1,4,0,6,1,4,1,3,1,8,2,3,2,5,2,8,2,2,5,4,4,6,6,3,5,5,2,5,9,4,
%T 7,3,7,5,4,5,9,4,5,6,3,8,7,5,5,11,5,7,4,6,3,6,5,6,5,6,5,6,3,4,6,3,5,4,
%U 5,7,6,4,5,5,4,3,9,6,4,5,4,6,4,3,5,8,3,7,9,10,8,7,2,8,3,6,6,8,8,3
%N Number of ways to write n = k + m (k, m > 0) with 2^k + prime(m) prime.
%C Conjecture: a(n) > 0 except for n = 1, 2, 7.
%C We have verified this for n up to 3*10^7. For n = 15687374, the least positive integer k with 2^k + prime(n-k) prime is 51299. For n = 28117716, the least positive integer k with 2^k + prime(n-k) prime is 81539.
%H Zhi-Wei Sun, <a href="/A233150/b233150.txt">Table of n, a(n) for n = 1..6000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n + b*n modulo m</a>, preprint, 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], 2014-2016.
%e a(9) = 1 since 9 = 7 + 2 with 2^7 + prime(2) = 128 + 3 = 131 prime.
%e a(13) = 1 since 13 = 3 + 10 with 2^3 + prime(10) = 8 + 29 = 37 prime.
%e a(588) = 1 since 588 = 66 + 522 with 2^{66} + prime(522) = 2^{66} + 3739 = 73786976294838210203 prime.
%e a(1012) = 1 since 1012 = 317 + 695 with 2^{317} + prime(695) = 2^{317} + 5231 prime.
%t a[n_]:=Sum[If[PrimeQ[2^k+Prime[n-k]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000079, A231201, A231557, A231577, A233183, A233204, A233206, A233296.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Dec 05 2013
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