A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).

5984, 7424, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655, 116864, 120015, 121904, 122264, 126224

Offset

1,1

Comments

Terms k such that both k and k+1 are primitive Zumkeller numbers (A180332) are 82004, 84524, 158235, 516704, 2921535, 5801984, ... (A361934).

There are infinitely many such k as proven by Somu et al. (2023). - Duc Van Khanh Tran, Dec 07 2023

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Giovanni Resta)

Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some results on Zumkeller numbers, arXiv:2310.14149 [math.NT], 2023.

Mathematica

zumkellerQ[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]]; zq1 = False; s = {}; Do[zq2 = zumkellerQ[n]; If[zq1 && zq2, AppendTo[s, n - 1]]; zq1 = zq2, {n, 2, 10^5}]; s (* after T. D. Noe at A083207 *)

Prog

(Python)
from itertools import count, islice
from sympy import divisors
def A328327_gen(startvalue=1): # generator of terms >= startvalue
    m = -1
    for n in count(max(startvalue, 1)):
        d = divisors(n)
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for _ in range(s2+1)] for _ in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y, s2+1):
                    z[i][j] = max(z[i-1][j], z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    if m == n-1:
                        yield m
                    m = n
                    break
A328327_list = list(islice(A328327_gen(), 5)) # Chai Wah Wu, Feb 13 2023

Crossrefs

Cf. A083207, A180332, A361934.

Subsequence of A096399.

Sequence in context: A157666 A176374 A282191 * A364861 A237011 A274362

Adjacent sequences: A328324 A328325 A328326 * A328328 A328329 A328330

Keyword

nonn

Author

Amiram Eldar, Oct 12 2019

Status

approved