A335220 Exponential Zumkeller numbers (A335218) whose set of exponential divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

36, 900, 3600, 22500, 44100, 176400, 705600, 1587600, 4410000, 5336100, 21344400

Offset: 1

Author

Amiram Eldar, May 27 2020

Keywords

• nonn
• more

Comments

The corresponding record values are 1, 3, 4, 6, 83, 2920, 81080, 254566, 344022, 487267, 4580715031, ...

Examples

			36 is the first term since it is the least exponential Zumkeller number, and its exponential divisors, {6, 12, 18, 36}, can be partitioned in a single way: 6 + 12 + 18 = 36. The next exponential Zumkeller number with more than one partition is 900, whose nonunitary divisors, {30, 60, 90, 150, 180, 300, 450, 900}, can be partitioned in 3 ways: 30 + 60 + 90 + 150 + 300 + 450 = 180 + 900, 60 + 90 + 180 + 300 + 450 = 30 + 150 + 900, and 150 + 180 + 300 + 450 = 30 + 60 + 90 + 900.
		

Crossrefs

The exponential version of A083212.

Subsequence of A335218.

Cf. A335219.

Programs

Mathematica

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; eDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expDivQ[n, #] &]]; nways[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], 0, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]]/2]]; nwaysm = 0; s = {}; Do[nways1 = nways[n]; If[nways1 > nwaysm, nwaysm = nways1; AppendTo[s, n]], {n, 1, 23000}]; s