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

Author

Amiram Eldar, Oct 12 2019

Keywords

• nonn

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.

Crossrefs

Cf. A083207, A180332, A361934.

Subsequence of A096399.

Programs

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 *)

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