A339035 - OEIS

%I #10 Dec 22 2020 17:28:49

%S 2,5,11,13,19,23,29,41,43,47,53,59,79,83,89,101,103,107,109,113,131,

%T 137,139,149,151,167,173,179,181,191,197,223,227,229,233,239,251,257,

%U 263,269,271,281,293,307,311,317,347,349,353,359,367,379,383,389,401,409,419,431,433,439,443

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

%C 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 Robert Israel, <a href="/A339035/b339035.txt">Table of n, a(n) for n = 1..10000</a>

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

%p Split:= proc(S, s, t) option remember;

%p   local m, Sp;

%p   if t = 0 then return true fi;

%p   if t > s then return false fi;

%p   m:= max(S);

%p   Sp:= S minus {m};

%p   (t >= m and procname(Sp,s-m,t-m)) or procname(Sp,s-m,t)

%p end proc:

%p isZumkeller:=  proc(n) local D,sigma; D:= numtheory:-divisors(n); sigma:= convert(D,`+`); sigma::even and

%p Split(D, sigma, sigma/2) end proc:

%p select(n -> isprime(n) and isZumkeller(2*(n+1)), [2,seq(i,i=3..1000)]); # _Robert Israel_, Dec 22 2020

%t 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 Select[Prime[Range[100]],zumkellerQ[2*(#+1)]&] (* zumkellerQ by _Jean-François Alcover_ at A320518 *)

%Y Cf. A000040, A083207, A320518 (subsequence).

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Nov 20 2020