%I #16 Jul 20 2022 11:15:49
%S 27720,30240,32760,50400,55440,60480,65520,75600,83160,85680,90720,
%T 95760,98280,100800,105840,110880,115920,120120,120960,128520,131040,
%U 138600,143640,151200,163800,166320,171360,180180,181440,184800,191520
%N Numbers n whose divisors can be partitioned into four disjoint sets whose sums are all sigma(n)/4.
%C Subsequence of A023198 (numbers n such that sigma(n) >= 4n).
%H Farid Jokar, <a href="https://arxiv.org/abs/2207.09053">On k-layered numbers</a>, arXiv:2207.09053 [math.NT], 2022.
%e Number 27720 is in the sequence because sigma(27720)/4 = 28080 = 360 + 27720 = 20 + 60 + 280 + 2310 + 4620 + 6930 + 13860 = 9 + 30 + 420 + 1540 + 1980 + 2772 + 3080 + 3465 + 5544 + 9240 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 12 + 14 + 15 + 18 + 21 + 22 + 24 + 28 + 33 + 35 + 36 + 40 + 42 + 44 + 45 + 55 + 56 + 63 + 66 + 70+ 72 + 77 + 84 + 88 + 90 + 99 + 105 + 110 + 120 + 126 + 132 + 140 + 154 + 165 + 168 + 180 + 198 + 210 + 220 + 231 + 252 + 264+ 308 + 315 + 330 + 385 + 396 + 440 + 462 + 495 + 504 + 616 + 630 + 660 + 693 + 770 + 792 + 840 + 924 + 990 + 1155 + 1260 + 1320 + 1386 + 1848 + 2520 + 3960 (summands are all divisors of 27720).
%p with(numtheory);with(combstruct);
%p A204831:=proc(i)
%p local S,R,Stop,Comb,c,d,k,m,n,s;
%p for n from 1 to i do
%p s:=sigma(n); c:=op(divisors(n));
%p if (modp(s,4)=0 and 4*n<=s) then
%p S:=1/4*s-n; R:=select(m->m<=S,[c]); Stop:=false;
%p Comb:=iterstructs(Combination(R));
%p while not (finished(Comb) or Stop) do
%p Stop:=add(d,d=nextstruct(Comb))=S;
%p od;
%p if Stop then print(n); fi;
%p fi;
%p od;
%p end:
%p A204831(100000); # _Paolo P. Lava_, Jan 24 2012
%Y Cf. A083207 (Zumkeller numbers--numbers n whose divisors can be partitioned into two disjoint sets whose sums are both sigma(n)/2), A204830 (numbers n whose divisors can be partitioned into three disjoint sets whose sums are all sigma(n)/3).
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Jan 22 2012