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A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

%I A233390 #25 Aug 05 2019 09:27:08
%S A233390 0,1,1,1,2,1,2,2,2,1,4,4,4,2,3,2,3,3,7,4,4,5,3,4,5,5,5,6,7,6,5,4,4,9,
%T A233390 3,6,6,5,4,7,1,4,5,6,9,6,8,6,8,4,5,8,7,4,3,4,7,6,6,3,6,5,6,4,6,8,7,8,
%U A233390 4,5,3,6,7,7,3,10,7,5,6,10,4,8,4,6,7,6,8,10,4,6,8,9,5,6,5,7,13,5,5,6
%N A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).
%C A233390 Conjecture: a(n) > 0 for all n > 1.
%C A233390 We have verified this for n up to 150000. For n = 124669, the least positive integer k with 2^k - 1 + q(n-k) prime is 13413.
%H A233390 Zhi-Wei Sun, <a href="/A233390/b233390.txt">Table of n, a(n) for n = 1..8000</a>
%H A233390 Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%H A233390 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2017.
%e A233390 a(6) = 1 since 2^2 - 1 + q(4) = 3 + 2 = 5 is prime.
%e A233390 a(10) = 1 since 2^4 - 1 + q(6) = 15 + 4 = 19 is prime.
%e A233390 a(41) = 1 since 2^{16} - 1 + q(25) = 65535 + 142 = 65677 is prime.
%e A233390 a(127) = 1 since 2^{21} - 1 + q(106) = 2097151 + 728260 = 2825411 is prime.
%e A233390 a(153) = 1 since 2^{70} - 1 + q(83) = 1180591620717411303423 + 101698 = 1180591620717411405121 is prime.
%e A233390 a(164) = 1 since 2^{26} - 1 + q(138) = 67108863 + 8334326 = 75443189 is prime.
%t A233390 a[n_]:=Sum[If[PrimeQ[2^k-1+PartitionsQ[n-k]],1,0],{k,1,n-1}]
%t A233390 Table[a[n],{n,1,100}]
%Y A233390 Cf. A000009, A000040, A000225, A233307, A233346, A233359, A233360, A233393, A233417, A233439.
%K A233390 nonn
%O A233390 1,5
%A A233390 _Zhi-Wei Sun_, Dec 08 2013