A320518 -id:A320518 - cp's OEIS frontend

A337399 Numbers k such that sigma(k) is a Zumkeller number (A083207).

5, 6, 11, 12, 14, 15, 19, 20, 23, 24, 26, 27, 28, 29, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 47, 53, 54, 56, 57, 58, 59, 60, 62, 63, 65, 68, 69, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 99, 101, 102, 103

Offset: 1

Ivan N. Ianakiev, Aug 26 2020

Sequence contains many semiprimes of the form p(m)*p(m+1). Only 6 of the first 200 semiprimes of this form are not terms, those where m is in {15, 37, 99, 100, 121, 197}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A083207, A320518 (subsequence).

zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]]; Select[Range[200],zQ[DivisorSigma[1,#]]&] (* code by T. D. Noe at A083207 *)

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

Original entry on oeis.org

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

Ivan N. Ianakiev, Nov 20 2020

nonn

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.

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

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

Cf. A000040, A083207, A320518 (subsequence).

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

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 *)

cp's OEIS Frontend

A337399 Numbers k such that sigma(k) is a Zumkeller number (A083207).

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