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

2, 5, 11, 13, 19, 23, 29, 41, 43, 47, 53, 59, 79, 83, 89, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 167, 173, 179, 181, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 349, 353, 359, 367, 379, 383, 389, 401, 409, 419, 431, 433, 439, 443

Offset: 1

Author

Ivan N. Ianakiev, Nov 20 2020

Keywords

• nonn

Comments

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.

Examples

			13 is prime and 2*(13+1) = 28 is Zumkeller, so 13 is a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A083207, A320518 (subsequence).

Programs

Maple

Split:= proc(S, s, t) option remember;
  local m, Sp;
  if t = 0 then return true fi;
  if t > s then return false fi;
  m:= max(S);
  Sp:= S minus {m};
  (t >= m and procname(Sp,s-m,t-m)) or procname(Sp,s-m,t)
end proc:
isZumkeller:=  proc(n) local D,sigma; D:= numtheory:-divisors(n); sigma:= convert(D,`+`); sigma::even and
Split(D, sigma, sigma/2) end proc:
select(n -> isprime(n) and isZumkeller(2*(n+1)), [2,seq(i,i=3..1000)]); # Robert Israel, Dec 22 2020

Mathematica

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]];
Select[Prime[Range[100]],zumkellerQ[2*(#+1)]&] (* zumkellerQ by Jean-François Alcover at A320518 *)