A335218 Exponential Zumkeller numbers: numbers whose exponential divisors can be partitioned into two disjoint subsets of equal sum.

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3600, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4356, 4500, 4572, 4716, 4788, 4900

Offset

1,1

Comments

First differs from A318100 at n = 49: 4900 is a term that is not an exponential pseudoperfect number.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

36 is a term since its exponential divisors, {6, 12, 18, 36}, can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

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, #] &]]; ezQ[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^4], ezQ]

Crossrefs

The exponential version of A083207.

Subsequence of A129575.

A054979 is a subsequence.

Cf. A318100, A322791, A323343.

Sequence in context: A348962 A127657 A318100 * A321145 A054979 A335219

Adjacent sequences: A335215 A335216 A335217 * A335219 A335220 A335221

Keyword

nonn

Author

Amiram Eldar, May 27 2020

Status

approved