A335144 - OEIS

%I #11 Jun 10 2020 04:19:25

%S 24,96,180,216,240,360,480,720,1080,1440,2160,2880,4320,5040,7560,

%T 10080,15120,20160,25200,30240,45360,50400,60480,75600,100800,110880,

%U 151200,221760,277200,302400,332640,453600,498960,554400,665280,831600,1108800,1330560

%N Nonunitary Zumkeller numbers (A335142) whose set of nonunitary divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

%C The corresponding record values are 1, 3, 7, 13, 17, 102, 140, ... (see the link for more values).

%H Amiram Eldar, <a href="/A335144/a335144.txt">Table of n, a(n), number of ways for n = 1..38</a>

%e 24 is the first term since it is the least nonunitary Zumkeller number, and its nonunitary divisors, {2, 4, 6, 12}, can be partitioned in a single way: 2 + 4 + 6 = 12. The next nonunitary Zumkeller number with more than one partition is 96, whose nonunitary divisors, {2, 4, 6, 8, 12, 16, 24, 48}, can be partitioned in 3 ways: 2 + 4 + 6 + 8 + 16 + 24 = 12 + 48, 2 + 6 + 12 + 16 + 24 = 4 + 8 + 48, and 8 + 12 + 16 + 24 = 2 + 4 + 6 + 48.

%t nuz[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] > 1 &], sum, x}, sum = Plus @@ d; If[sum < 1 || OddQ[sum], 0, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]]/2]]; nuzm = 0; s = {}; Do[nuz1 = nuz[n]; If[nuz1 > nuzm, nuzm = nuz1; AppendTo[s, n]], {n, 1, 8000}]; s

%Y The nonunitary version of A083212.

%Y Subsequence of A335142.

%Y Cf. A335143.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 25 2020