A391089 Primitive exponential unitary Zumkeller numbers: powerful numbers whose exponential unitary divisors can be partitioned into two disjoint subsets of equal sum.

36, 900, 1764, 1800, 2700, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844, 213444

Offset

1,1

Comments

Any term of A391088 is of the form k*m where k is a term of this sequence and m is a squarefree number coprime to k. Therefore, A391088 can be generated from this sequence by multiplying terms with coprime squarefree numbers, and the asymptotic density of A391088 can be evaluated from the terms of this sequence (see the Comments section of A391088).

The least odd term is a(1458) = A321147(1) = 225450225 = (3 * 5 * 7 * 11 * 13)^2, which is the least odd exponential abundant number.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10127 (terms below 10^10)

Formula

36 = 2^2 * 3^2 is a term since it is a powerful number, and its exponential unitary divisors, {6, 12, 18, 36}, can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

Mathematica

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
seq[max_] := Select[pows[max], euzQ]; seq[200000] (* using the function "euzQ" from A391088 *)

Crossrefs

Intersection of A001694 and A391088.

Subsequence of A383694.

Cf. A005117, A361255.

Sequence in context: A169836 A391087 A391143 * A335220 A248108 A233003

Adjacent sequences: A391086 A391087 A391088 * A391090 A391091 A391092

Keyword

nonn

Author

Amiram Eldar, Nov 28 2025

Status

approved