A384515 Unitary s-Zumkeller numbers.

60, 140, 420, 660, 1224, 1820, 2660, 2820, 4620, 5460, 7140, 7980, 8580, 9660, 11220, 12180, 12540, 13020, 13260, 13580, 13860, 14140, 14420, 14820, 15180, 15540, 16380, 17220, 17940, 18060, 18200, 19140, 19380, 19740, 20020, 20460, 22260, 22620, 23460, 24180, 24420

Offset: 1

Author

Ivan N. Ianakiev, Jun 01 2025

Keywords

• nonn

Comments

A positive integer k is called a unitary s-Zumkeller number if the set D of proper positive unitary divisors of k can be partitioned as {A,B}, such that the sum of the squares of the elements of A equals the sum of the squares of the elements of B equals (sigma2U(k) - k^2)/2, where sigma2U(k) is the sum of the squares of the unitary divisors of k.

All terms are even numbers.

Examples

			The set of proper unitary divisors of 60 is A = {1,3,4,5,12,15,20}. The set of their squares is B= {1,9,16,25,144,225,400} and the sum of its elements is 820. B = {1,9,400} union {16,25,144,225}, which makes 60 a term of the present sequence.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• David A. Corneth, PARI program.
• Jayanta Kalita and Helen K. Saikia, A Study On Generalized Zumkeller Numbers, INTEGERS, Volume 24 (2024), A95, pp. 7-10.

Crossrefs

Cf. A034448, A077610, A083207, A290466.

Programs

Mathematica

(* Naive code to illustrate the terms found by Kalita and Saikia *)
uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];
sigma2UDiv[n_]:=Total[uDiv[n]^2]; propUDiv[n_]:=uDiv[n]//Most;
subsetsPropUDivQ[1]:=False; subsetsPropUDivQ[n_]:=
Select[Subsets[propUDiv[n]],Total[Flatten[#]^2]==(sigma2UDiv[n]-n^2)/2&]!={};
Select[Range[1820],subsetsPropUDivQ[#]&]
(* or *)
q[n_] := Module[{d = Select[Divisors[n], # < n && CoprimeQ[#, n/#] &]^2, sum, x}, sum = Total[d]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2, 1000], q] (* Amiram Eldar, Jun 01 2025 *)

PARI

\\ See Corneth link

Extensions

a(7)-a(16) from Amiram Eldar, Jun 01 2025

More terms from David A. Corneth, Jun 01 2025