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A384515 Unitary s-Zumkeller numbers.

%I A384515 #25 Jun 07 2025 18:42:09
%S A384515 60,140,420,660,1224,1820,2660,2820,4620,5460,7140,7980,8580,9660,
%T A384515 11220,12180,12540,13020,13260,13580,13860,14140,14420,14820,15180,
%U A384515 15540,16380,17220,17940,18060,18200,19140,19380,19740,20020,20460,22260,22620,23460,24180,24420
%N A384515 Unitary s-Zumkeller numbers.
%C A384515 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.
%C A384515 All terms are even numbers.
%H A384515 David A. Corneth, <a href="/A384515/b384515.txt">Table of n, a(n) for n = 1..10000</a>
%H A384515 David A. Corneth, <a href="/A384515/a384515.gp.txt">PARI program</a>.
%H A384515 Jayanta Kalita and Helen K. Saikia, <a href="https://math.colgate.edu/~integers/y95/y95.pdf">A Study On Generalized Zumkeller Numbers</a>, INTEGERS, Volume 24 (2024), A95, pp. 7-10.
%e A384515 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.
%t A384515 (* Naive code to illustrate the terms found by Kalita and Saikia *)
%t A384515 uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];
%t A384515 sigma2UDiv[n_]:=Total[uDiv[n]^2]; propUDiv[n_]:=uDiv[n]//Most;
%t A384515 subsetsPropUDivQ[1]:=False; subsetsPropUDivQ[n_]:=
%t A384515 Select[Subsets[propUDiv[n]],Total[Flatten[#]^2]==(sigma2UDiv[n]-n^2)/2&]!={};
%t A384515 Select[Range[1820],subsetsPropUDivQ[#]&]
%t A384515 (* or *)
%t A384515 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 *)
%o A384515 (PARI) \\ See Corneth link
%Y A384515 Cf. A034448, A077610, A083207, A290466.
%K A384515 nonn
%O A384515 1,1
%A A384515 _Ivan N. Ianakiev_, Jun 01 2025
%E A384515 a(7)-a(16) from _Amiram Eldar_, Jun 01 2025
%E A384515 More terms from _David A. Corneth_, Jun 01 2025