A391090 Exponential Zumkeller numbers that are not exponential unitary Zumkeller numbers.

3600, 7056, 8100, 17424, 25200, 35280, 39600, 46800, 56700, 61200, 68400, 77616, 82800, 87120, 89100, 91728, 104400, 105300, 111600, 119952, 121968, 133200, 134064, 137700, 147600, 153900, 154800, 162288, 169200, 186300, 190800, 204624, 212400, 218736, 219600, 226512

Offset

1,1

Comments

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 (A391091) 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(A391091(n)) = 0.0001626..., 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(6196002) = A321147(199) = 38101088025 = (3 * 5 * 7 * 11 * 13^2)^2.

Links

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

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

Example

3600 is a term since its exponential divisors, {30, 60, 90, 150, 180, 240, 300, 450, 720, 900, 1200, 3600}, can be partitioned into 2 disjoint sets whose sum is equal: 60 + 300 + 3600 = 30 + 90 + 150 + 180 + 240 + 450 + 720 + 900 + 1200, but its exponential unitary divisors, {30, 90, 150, 240, 450, 720, 1200, 3600}, cannot be partitioned in this way.

Mathematica

seq[max_] := Select[Range[max], ezQ[#] && !euzQ[#] &]; seq[230000] (* using the functions "ezQ" from A335218 and "euzQ" from A391088 *)

Crossrefs

Equals A335218 \ A391088.

A391091 is a subsequence.

Cf. A001694, A005117, A321147, A322791, A361255.

Sequence in context: A348521 A175752 A179746 * A391091 A391085 A391086

Adjacent sequences: A391087 A391088 A391089 * A391091 A391092 A391093

Keyword

nonn

Author

Amiram Eldar, Nov 28 2025

Status

approved