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.

From Amiram Eldar, Nov 28 2025: (Start)

Also called "e-Zumkeller numbers" by Kalita and Saikia (2025).

The set of exponential divisors of a number k is equal to the set of exponential divisors of the powerful part of k (A057521), each multiplied by the powerfree part of k (A055231). Therefore, if k is a term and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms of this sequence (A391087) are the powerful (A001694) terms. All the terms are of the form k*m where k is primitive and m is a squarefree number coprime to k.

The asymptotic density of this sequence is Sum_{n>=1} f(A391087(n)) = 0.0097377..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).

All the odd terms are exponential abundant numbers (A129575), since there are no odd e-perfect number (A054979), as proved by Straus and Subbarao (1974).

The least odd term is a(2195355) = A321147(1) = 225450225 = (3 * 5 * 7 * 11 * 13)^2, which is the least odd exponential abundant number.

Are all the odd exponential abundant numbers (A321147) terms in this sequence? If not, the least term in A321147 that is not a term in this sequence is larger than 10^18. (End)

Links

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

Jayanta Kalita and Helen K. Saikia, e-Zumkeller Numbers and e-Unitary Zumkeller Numbers, Palestine Journal of Mathematics, Vol. 14, No. 3 (2025), pp. 684-691.

E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J., Vol. 41, No. 2 (1974), pp. 465-471.

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 the union of A054979 and A129575.

Subsequences: A054979, A391087.

Cf. A001694, A005117, A055231, A057521, A318100, A321147, A322791, A323343.

Similar sequences: A290466, A335197, A335142, A335215, A339979, A348527.

Sequence in context: A348962 A127657 A318100 * A391088 A321145 A054979

Adjacent sequences: A335215 A335216 A335217 * A335219 A335220 A335221

Keyword

nonn

Author

Amiram Eldar, May 27 2020

Status

approved