cp's OEIS Frontend

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

%I A328327 #36 Dec 07 2023 16:10:44
%S A328327 5984,7424,21735,21944,26144,27404,39375,43064,49664,56924,58695,
%T A328327 61424,69615,70784,76544,77175,79695,81080,81675,82004,84524,84644,
%U A328327 89775,91664,98175,103455,104895,106784,109395,111824,116655,116864,120015,121904,122264,126224
%N A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).
%C A328327 Terms k such that both k and k+1 are primitive Zumkeller numbers (A180332) are 82004, 84524, 158235, 516704, 2921535, 5801984, ... (A361934).
%C A328327 There are infinitely many such k as proven by Somu et al. (2023). - _Duc Van Khanh Tran_, Dec 07 2023
%H A328327 Amiram Eldar, <a href="/A328327/b328327.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Giovanni Resta)
%H A328327 Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, <a href="https://arxiv.org/abs/2310.14149">Some results on Zumkeller numbers</a>, arXiv:2310.14149 [math.NT], 2023.
%t A328327 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 *)
%o A328327 (Python)
%o A328327 from itertools import count, islice
%o A328327 from sympy import divisors
%o A328327 def A328327_gen(startvalue=1): # generator of terms >= startvalue
%o A328327     m = -1
%o A328327     for n in count(max(startvalue,1)):
%o A328327         d = divisors(n)
%o A328327         s = sum(d)
%o A328327         if s&1^1 and n<<1<=s:
%o A328327             d = d[:-1]
%o A328327             s2, ld = (s>>1)-n, len(d)
%o A328327             z = [[0 for _ in range(s2+1)] for _ in range(ld+1)]
%o A328327             for i in range(1, ld+1):
%o A328327                 y = min(d[i-1], s2+1)
%o A328327                 z[i][:y] = z[i-1][:y]
%o A328327                 for j in range(y,s2+1):
%o A328327                     z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
%o A328327                 if z[i][s2] == s2:
%o A328327                     if m == n-1:
%o A328327                         yield m
%o A328327                     m = n
%o A328327                     break
%o A328327 A328327_list = list(islice(A328327_gen(),5)) # _Chai Wah Wu_, Feb 13 2023
%Y A328327 Cf. A083207, A180332, A361934.
%Y A328327 Subsequence of A096399.
%K A328327 nonn
%O A328327 1,1
%A A328327 _Amiram Eldar_, Oct 12 2019