A390748 Zumkeller numbers that are not oblong numbers.

24, 28, 40, 48, 54, 60, 66, 70, 78, 80, 84, 88, 96, 102, 104, 108, 112, 114, 120, 126, 138, 140, 150, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 216, 220, 222, 224, 228, 234, 246, 252, 258, 260, 264, 270, 276, 280, 282, 294, 300, 304, 308, 312, 318, 320, 330, 336, 340, 348, 350, 352, 354, 360

Offset

1,1

Comments

It seems that the maximum first difference is 18.

If k is a term, p is a prime coprime to k and p is not equal to k-1 or k+1, then k*p is a term.

945 is the first odd term.

Except for 6, all perfect numbers (A000396) are terms.

Links

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

Example

Let D be the set of the divisors of 24, i.e. {1,2,3,4,6,8,12,24}. D = {1,2,3,24} union {4,6,8,12}. So 24 is a Zumkeller number, but it is not of the form n*(n+1). Therefore, 24 is a term.

Mathematica

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]]; (* After T. D. Noe at A083207 *)
oQ[n_] := Module[{s=Floor[Sqrt[n]]}, n==s*(s+1)]; Select[Range[360], !oQ[#]&&zQ[#]&]

Prog

(SageMath)
def is_A390748(n) -> bool:
    if is_square(4*n + 1): return False
    s = sigma(n)
    if not (2.divides(s) and n*2 <= s): return False
    S = s // 2 - n
    R = (m for m in divisors(n) if m <= S)
    return any(sum(c) == S for c in Combinations(R))
lim = 360; print([n for n in (1..lim) if is_A390748(n)])  # Peter Luschny, Nov 23 2025

Crossrefs

Intersection of A083207 and A078358.

Set difference of A083207 and A002378.

Cf. A000396.

Sequence in context: A268540 A030500 A107406 * A206261 A217441 A045668

Adjacent sequences: A390745 A390746 A390747 * A390749 A390750 A390751

Keyword

nonn

Author

Ivan N. Ianakiev, Nov 17 2025

Status

approved