A335144 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.

24, 96, 180, 216, 240, 360, 480, 720, 1080, 1440, 2160, 2880, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 45360, 50400, 60480, 75600, 100800, 110880, 151200, 221760, 277200, 302400, 332640, 453600, 498960, 554400, 665280, 831600, 1108800, 1330560

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..38.

Amiram Eldar, Table of n, a(n), number of ways for n = 1..38

Example

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.

Mathematica

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

Crossrefs

The nonunitary version of A083212.

Subsequence of A335142.

Cf. A335143.

Sequence in context: A057103 A055669 A370846 * A209432 A195824 A183009

Adjacent sequences: A335141 A335142 A335143 * A335145 A335146 A335147

Keyword

nonn

Author

Amiram Eldar, May 25 2020

Status

approved