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

1,1

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

Cf. A000040, A083207, A320518 (subsequence).

Sequence in context: A045360 A001915 A127437 * A084792 A109640 A191048

Adjacent sequences: A339032 A339033 A339034 * A339036 A339037 A339038

Keyword

nonn

Author

Ivan N. Ianakiev, Nov 20 2020

Status

approved