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A339035 k is prime and 2*(k+1) is Zumkeller.

%I A339035 #10 Dec 22 2020 17:28:49
%S A339035 2,5,11,13,19,23,29,41,43,47,53,59,79,83,89,101,103,107,109,113,131,
%T A339035 137,139,149,151,167,173,179,181,191,197,223,227,229,233,239,251,257,
%U A339035 263,269,271,281,293,307,311,317,347,349,353,359,367,379,383,389,401,409,419,431,433,439,443
%N A339035 k is prime and 2*(k+1) is Zumkeller.
%C A339035 This is a supersequence of A320518. If k+1 is Zumkeller, then 2*(k+1) is also Zumkeller (see my Lemma 1 at the Links section of A002182), which makes all terms of A320518 terms of this sequence. The reverse is not true, so this sequence contains terms that are not terms of A320518, such as 2,13,43, etc.
%H A339035 Robert Israel, <a href="/A339035/b339035.txt">Table of n, a(n) for n = 1..10000</a>
%e A339035 13 is prime and 2*(13+1) = 28 is Zumkeller, so 13 is a term.
%p A339035 Split:= proc(S, s, t) option remember;
%p A339035   local m, Sp;
%p A339035   if t = 0 then return true fi;
%p A339035   if t > s then return false fi;
%p A339035   m:= max(S);
%p A339035   Sp:= S minus {m};
%p A339035   (t >= m and procname(Sp,s-m,t-m)) or procname(Sp,s-m,t)
%p A339035 end proc:
%p A339035 isZumkeller:=  proc(n) local D,sigma; D:= numtheory:-divisors(n); sigma:= convert(D,`+`); sigma::even and
%p A339035 Split(D, sigma, sigma/2) end proc:
%p A339035 select(n -> isprime(n) and isZumkeller(2*(n+1)), [2,seq(i,i=3..1000)]); # _Robert Israel_, Dec 22 2020
%t A339035 zumkellerQ[n_]:=Module[{d=Divisors[n],ds,x},ds=Total[d];If[OddQ[ds],False,SeriesCoefficient[Product[1+x^i,{i,d}],{x,0,ds/2}]>0]];
%t A339035 Select[Prime[Range[100]],zumkellerQ[2*(#+1)]&] (* zumkellerQ by _Jean-François Alcover_ at A320518 *)
%Y A339035 Cf. A000040, A083207, A320518 (subsequence).
%K A339035 nonn
%O A339035 1,1
%A A339035 _Ivan N. Ianakiev_, Nov 20 2020