A331371 - OEIS

%I #15 Apr 04 2023 07:43:39

%S 224,440,1224,2024,3968,5624,11024,18224,35720,38024,50624,53360,

%T 65024,74528,81224,140624,148224,159200,164024,184040,189224,194480,

%U 207024,216224,233288,245024,314720,354024,370880,378224,416024,423800,442224,455624,497024,511224

%N Numbers k such that k and k+1 are both half-Zumkeller numbers (A246198).

%H Amiram Eldar, <a href="/A331371/b331371.txt">Table of n, a(n) for n = 1..10000</a>

%e 224 is a term since both 224 and 225 are half-Zumkeller numbers: the proper divisors of 224 are {1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112} and 1 + 2 + 4 + 7 + 8 + 14 + 16 + 32 + 56 = 28 + 112, and the proper divisors of 225 are {1, 3, 5, 9, 15, 25, 45, 75} and 1 + 3 + 15 + 25 + 45 = 5 + 9 + 75.

%t hzQ[n_] := Module[{d = Most @ Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; hzq1 = False; s = {}; Do[hzq2 = hzQ[n]; If[hzq1 && hzq2, AppendTo[s, n - 1]]; hzq1 = hzq2, {n, 2, 6000}]; s

%Y Cf. A246198, A246199, A328327.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 03 2020