%I #11 Dec 30 2020 18:31:45

%S 1,9,11,15,34,36,43,80,152,159,168,200,205,354,402,957,1898,2519,2729,

%T 2932,3075,3740,4985,5839,7911,9868,10210,24624,27735,31553,37190

%N Positive integers m with 2^m*p(m) + 1 prime, where p(.) is the partition function (A000041).

%C According to the conjecture in A236389, this sequence should have infinitely many terms.

%C The prime 2^(a(31))*p(a(31)) + 1 = 2^(37190)*p(37190) + 1 has 11405 decimal digits.

%H Zhi-Wei Sun, <a href="/A236390/b236390.txt">Table of n, a(n) for n = 1..31</a>

%e a(1) = 1 since 2^1*p(1) + 1 = 2*1 + 1 = 3 is prime.

%t q[n_]:=PrimeQ[2^n*PartitionsP[n]+1]

%t n=0;Do[If[q[m],n=n+1;Print[n," ",m]],{m,1,10000}]

%t Select[Range[40000],PrimeQ[2^# PartitionsP[#]+1]&] (* _Harvey P. Dale_, Dec 30 2020 *)

%Y Cf. A000040, A000041, A236389.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jan 24 2014