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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. 0
36, 900, 3600, 22500, 44100, 176400, 705600, 1587600, 4410000, 5336100, 21344400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..11.

EXAMPLE

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.

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

CROSSREFS

The exponential version of A083212.

Subsequence of A335218.

Cf. A335219.

Sequence in context: A001812 A229680 A169836 * A248108 A233003 A075916

Adjacent sequences:  A335217 A335218 A335219 * A335221 A335222 A335223

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, May 27 2020

STATUS

approved

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Last modified September 22 22:24 EDT 2020. Contains 337291 sequences. (Running on oeis4.)