A335219 Exponential Zumkeller numbers (A335218) whose set of exponential divisors can be partitioned into two disjoint sets of equal sum in a single way.

36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4572, 4716, 4788, 4900, 4932

Offset: 1

Author

Amiram Eldar, May 27 2020

Keywords

• nonn

Comments

Differs from A054979 first at a(44), since 4900 is in this sequence but not in A054979. - R. J. Mathar, Jun 02 2020

Examples

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

Links

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

Crossrefs

The exponential of A083209.

Subsequence of A335218.

Cf. A335143, A335199, A335202.

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, #] &]]; 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]] == 2]]; Select[Range[10^4], ezQ]