login
A339035
k is prime and 2*(k+1) is Zumkeller.
1
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
1,1
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
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
Cf. A000040, A083207, A320518 (subsequence).
Sequence in context: A045360 A001915 A127437 * A084792 A109640 A191048
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, Nov 20 2020
STATUS
approved