OFFSET
1,1
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
13 is prime and 2*(13+1) = 28 is Zumkeller, so 13 is a term.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Nov 20 2020
STATUS
approved